The human eye focusing is resolving all the possible detail it can from a scene that is sharp and not distorted. The details of exactly how your brain forms an image from what your eye does is extremely complex, but the basics are : sharp initial image can be focused on to produce sharp image.

The blurry photo cannot be sharply resolved in that way because (surprise) the data is simply not there to make it focused. In effect the "sharpest image" of a blurry scene is a blurry image. It's a faithful rendering of the scene.

The process for (trying) to deblur a blurry image is called convolution. This is not the same or the reverse of focusing on a sharp scene to produce a sharp image. They work differently. A lens like the eye cannot do that complex operation. Even that deblurring process (which you can see done in many photos nowadays by software) is not 100% and it effectively makes "educated guesses" about what the scene originally looked like – guesses guided by good math and physics but still guesses (I suppose "estimate" would be a fairer word).

Part of this question should be straightened out in biology SE. In my knowledge the eye does not flex the retina to get an image.

What remains is why a lens cannot make a sharp image out of a blurred photo. It must be obvious that lenses cannot see sharp through a fog. It cannot undo the scattering of a billion randomly placed tiny water droplets. A lens can undo a blurry image due to defocus. This is because the light contains much more information than the image. A photo taken at defocus no longer contains this information.

Here's an experiment you can do. Hold a finger up a foot or so from your eye when in front of a window. If you focus on the finger, distant objects will be out of focus. You can then focus on those objects and they will be in focus. Your eye is adjusting first for the distance to your finger and then for the distance to the distant objects. Now if you look at a blurry photo print say on a wall, all you can do is focus on the print. If you try to focus beyond the print there is no information about the print beyond the wall. Ditto if you try to focus in front of the print. There is no distance from you at which the image is in focus, so you have nothing sharp to focus on. Only a real object can be focused by adjustment of your eye (assuming it's at an appropriate distance for you particular vision) because it is sharp to begin with.

Here's an explanation using geometric optics.

I'll replace the human eye, which has a quasirandom scattering of light sensors on a curved surface, with a digital camera that has a regular grid of sensors on a flat surface. This doesn't alter the problem in any essential way. Let's also not worry about color.

Say the camera is focused on a (monochrome) LCD monitor that is displaying some image. With an ideal lens and very careful alignment, the rays emitted by each pixel of the monitor will be focused onto a unique pixel of the camera sensor, and you will get a pixel-perfect reproduction of the original image.

Move the LCD monitor farther from the camera, so it's no longer in focus. Instead of using the lens equation to model this as you usually would, just use the fact that the lens focuses rays coming from points in the focal plane to points on the sensor. Those rays are still there, since the light passes through the focal plane on its way to the camera. The difference is that the rays emerging in different directions from a given point in the focal plane now come from different pixels of the monitor. This means that they have different brightnesses.

The camera sensor doesn't record the direction-dependent brightness of the light that hits it; it only records the total intensity (which becomes an average intensity once you normalize the overall exposure). If you take the (blurry) photograph recorded by the camera and display it on a monitor in the focal plane, you aren't accurately reproducing the light that was there before, because the monitor wrongly emits light of (roughly) the same brightness in all directions from each point. The refocusing operation that would have worked for the original light field doesn't work for this radically different light field.

So-called light-field cameras, which record the intensity of light arriving at the same point on the film from different directions separately, do exist. If you took the data recorded from such a camera, and displayed it on a monitor that could reproduce the light field it measured, then you actually could refocus your eye (or a standard camera) to see a sharp reproduction of what was on the original LCD monitor, whether or not the light-field camera was focused on it. (The notion of focus is largely meaningless for light-field cameras.)

The eye doesn't "unblur" a blurry image. There are excellent answers here already, but to help build intuition, it might be helpful to understand what software (like Photoshop) does when you tell it to artificially blur an image.

It essentially goes pixel by pixel, and combines (basically via a weighted average) a bunch of neighboring pixels into one. So for each pixel, the color/intensity information gets all scrambled up with that of surrounding pixels, and the image becomes blurry. The larger this "scrambling region" is, the blurrier it gets. This is what people mean when they say "information is lost" - there's no easy way to unscramble this, especially for very blurry images. The ability of the eye to focus is not one that's based on unscrambling; the eye (and our brain) cannot do that.

When it comes to seeing with an eye, or taking a photo with a camera, the mechanism is different, but the end result is more or less the same. You see objects because light bounces off of their entire lit surface in all directions, and some of those rays happen to go in the direction of your eye. When the lens is not focused on an object, the light bouncing off of some region of the surface (and at the edges, even light coming from behind the object) ends up in the same place on the retina/sensor, so as before, the information gets all scrambled up.

Now, when you focus, your eye doesn't take that information and unscramble it. Instead, it changes the curvature of the lens, so that the light rays are refracted at a different angle, eventually forming a clear image from new information carried by incoming "stream" of light rays. You can roughly think of it as frames in a movie: it's not that the eye/brain unscrambled the current frame, but rather that each new frame that comes in happens to be progressively more focused - thanks to the action of the lens.

When you take a blurry photo, the information recorded on it is already scrambled. When you look at it, it doesn't appear blurry because it's out of focus, but because it actually looks like that - that's how it's colored. When you look at a blurry picture, the rays bouncing off of different parts of it (or emitted, if on a screen) don't maintain their original relationships (they don't come from the same angles as before), so no matter what your eye does, it can't make it appear more focused.

I love this question! There is a subtle contradiction that you are hitting upon here and it is important to resolve it.

A camera and your eye both work in approximately the same way, and I will try to use some specific words to erase the difference between them. These are: light from some scene (a beach you are looking at, or an apple you are photographing) is directed through a tiny hole (the pupil of your eye, the aperture of the camera) to form an inverted image on a screen of sensors (rod and cone cells in your eye, the CCD grid in a modern camera, the silver halide crystals on older photographic film). In the process a lens is also used to bend the light on its way in; it is not essential to the function of the camera but makes it more useful. There may also be some notion of filtering that is not essential—black-and-white photography still exists, but your eye has different cells with different responsiveness to different colors of light and photography often tries to mirror this by creating components with similarly different responsiveness via color filters, but we can just imagine that we're talking about black-and-white photography and then layer on this detail of how the eye sees color at the end.

Just to get you thinking about holes

Now, the hole gets almost no love compared to the lens and the sensors, perhaps because it’s not something one spends money on whereas one spends hundreds of dollars on good camera lenses. But it is one of these most essential things which makes photography work. So if you are a human being, sitting on a beach, what makes this wave over here different from that wave over there, is that you have to look in different directions to see them. The camera needs to take this directional information and turn it into positional information, “which sensor at which position is picking up this wave, which other position is picking up that wave?”. And the hole is what allows that connection to take place. If you do not introduce the hole, then every part of the screen sees every wave and there is no notion of image, you just have a general “how bright is it outside today?” sensor but no ability to pick out the brightness in different directions. It is that you can draw a straight line from the sensor to the hole, and then continue that line outward to the scene. A hole, in other words, sits between two cones of directions that go through it, each direction going in towards the sensor corresponds perfectly to some direction going out towards the scene. And the correspondence is why the image is always inverted by 180 degrees as it goes through the hole, but we can rotate our film once we take the photograph and so this is no big deal.

Sources of blur, including the size of the hole

Now blur, comes from a couple different places.

1. The sensors have a finite size, and so they must aggregate together a bunch of different directions, averaging over the light that they see. This is called the resolution of the sensors, and it means that if I take an photo of something which is very far away without some sort of optical magnification, and then I try to blow up the details of that photo, some of those details are just missing because I didn’t have enough resolution.

2. As a practical detail, any camera has a characteristic response time. For my camera, I adjust it manually, it is called the “shutter speed.” Your rods and cones in your eye have this too. If the scene is changing, faster than this response time, then we are averaging over the different changed versions. This leads most notably to “motion blur” if I try to capture a ball in flight or so, but the higher my shutter speed, the more you will notice the jitter of my hands as I hold the camera as a sort of random motion blur on everything in the photograph. This is why you will see lots of tripods, and why professional photographers will often hold their arms very close to their sides, and so forth. So when you try to “blow up” that shot you may see my motion blur from my random jittering be more pronounced.

3. You may have noticed that nighttime photos are more noisy or staticky than daytime ones. This has to do with a sort of intrinsic noise in the sensors; they are built to expect a certain amount of light and if they do not get that then they get “underexposed” or “overexposed” and cannot use their full range of greys to represent the minute differences in light that they are seeing. Different film has different “ISO ratings”, what we might call more generically the sensors’ sensitivity. Your eye has to fine-tune its sensitivity, which is why it takes a while for you to obtain “night vision” after you turn off the lights; that failure to be able to make out details is a “blur” of sorts and once your eyes are adjusted you are often amazed to think that you had trouble walking through the room when now you can see everything quite plainly. You can make this mistake of under/over-exposure in daytime photography too, but you have to make more uncomfortable tradeoffs when you don’t have as much light to work with: in daylight you can set the shutter speed very low and make the aperture very narrow and still have a very bright image relative to the noise of your film; at night you have to choose either more noisy film or longer shutter speeds or larger apertures.

4. If the hole is too small, it can actually be smaller than light! Light has a characteristic size called its wavelength and when it passes through holes smaller than this, it does not behave in quite the same way. For visible light, though, you have to make the hole smaller than a human hair; our pupils are millimeters wide, so this is mostly a consideration for tiny pinhole cameras; let me describe it in a footnote ^[1].

5. Of course, there could be smudges on lenses and the like, which bend or absorb-and-reemit light so that light from the “wrong directions” hits a certain sensor. Finally:

6. The hole is of nonzero size.

This last one causes the sort of blur you are talking about in your question. If the hole were infinitely small, then there would actually never be any intrinsic reason for blur except perhaps for (4) above. And you know that something has to happen here because no hole is the same as having an arbitrarily big hole, so the largeness of the hole needs to somehow continuously decay our ability to do photography from “yes” at a hole of near-zero-size to “no” at a hole of, say, the size of the hole in my guitar.

The issue is simple: remember how for a single point we had a cone-of-directions producing those images on the screen of sensors? If the hole the light passes through is of nonzero size, then you can think of it as lots (say a million) of these cones-of-directions scattered across the virtual surface of the hole, each casting its own image onto the retina. The sensors on the screen, then, average all of these different images together. So each image has crisp lines, and where all of the images agree in projecting a line on the screen you also see crisp lines: but where those images differ, you see a blur as some of the images have that line a little to the left or to the right.

If you have no lens, and the scene is very far away, then the images are not very different, and so you do not notice this blur so much. Things that are closer to the camera are blurred more and more, as looking at the object from one point in the hole, it looks slightly different from how it looks from another point one millimeter beside it in the hole. We say that the pinhole camera is focused “at infinity” without a lens in front of it, and that the size of the hole determines something called “depth of field”, the smaller the pinhole the greater the depth of the field it can focus.^[2]

One can also think of this with that “guitar” analogy. If I look at my guitar, I can see a lot of the wood from the backboard through the sound hole. Imagine if that space is all covered with sensors, then from this distance I can see a lot of those sensors. If I put my eye right up next to the strings I can see entirely inside of the guitar, the whole backboard. If I move further away I see less and less until I can only see a chunk about the same size as the sound hole. What I see, are the sensors that can see me. The fewer sensors that can see me, the more crisp of an image I make on the back board. So that's why the smaller apertures make crisper images, if you prefer that explanation, and why by default you have the focus “at infinity” and there is therefore some characteristic distance for a pinhole camera where things that are “too close” get blurry.

How can a lens reduce blur?

So like I said, the lens only affects this last sort of blur, due to the images cast by different points of the hole being slightly different images, which the sensors must average over. How does it do that? By slightly warping different images from different locations.

See, when you are still at the pupil, you haven’t averaged yet. You still have all of the information: two dimensions of position combined with two dimensions of direction means that you have four dimensions of information available here to describe the light from a three-dimensional scene. It is only when everything gets averaged by the sensor, that all of this information is lost.

Cast a ray from some sensor $S$ to two different points in the hole, $H _1$ and $H_2$. With a pinhole camera these go along different lines, $overline{SH_1}$ and $overline{SH_2}$, and you know that they intersect at $S$ so they never intersect again. But if you put a lens in front of that pupil, you can change those directions again. Maybe now they converge again at some point $X$ out in the world. Or look at it from the reverse side, me-looking-at-my-guitar, it's like if one particular sensor on my guitar had been magnified so that it appeared to take up the whole sound hole. All the light from this area of the scene, going through that pupil, ends up at the same sensor. And now that is very much like the idea of the infinitely small pinhole: you get a crisp appearance for things at this fixed distance $L$ from the camera, because all of the images have been slightly distorted by the lens.

But it is only possible because the information has not been destroyed yet, it is still sitting in this 4-pack of 2 position coordinates in the pupil, and 2 direction coordinates from that point to the image surface. Once we average these images together we cannot uniquely specify what numbers led to them, any more than me telling you "I have three numbers which average to 5" could tell you "they are 4, 5, and 6" versus "they are 2, 6, and 7" versus "they are 4.9, 5.0, and 5.1." The inputs are now lost, and any choices I make are going to make various assumptions about distribution and approximation. This is why scientists laugh when a CSI television show says “enhance! enhance” to a photograph and it unblurs, we know that the more you enhance the more you just are confronted with the assumptions of your enhancement method and you start to see garbage which never appears in the “enhanced” photo (since the television crew started with a sharp image and blurred it—no assumptions needed there!—and is now playing this process in reverse for dramatic effect).

The introduction of a lens next to the hole (on the scene-side in a camera, on the image-side in your eye) creates a preferred distance $L$ for things to be in the scene. By controlling^[3] both $L$ and the depth of field, you can then create photographs of a woman sitting at a park bench, in full detail, with things closer and further away all blurred out of focus, and it can create a great intensity of attention on that woman and that bench. Those sorts of effects. And that is why our lenses are compound, we can adjust the effective focal length $L$ by adjusting the distance between two normal lenses. We also use lenses for overall distortion, say to include a wider angle in the shot than the sensors normally see; you will see this in apartment photographs all the time where the “straight lines,” carefully seen, curve a bit. And this gives a sense that the apartment is larger than it really is and also allows the photographer to take fewer shots as each shot captures more of the arrangements of walls and lights and cabinets than it otherwise would. Those sorts of distortions can generally be at least approximately undone. But the averaging that has happened at the sensor level, which manifests in the blur in the image, is intrinsic, so you are getting lots of answers saying that the information was “lost.”

The important thing to understand here is that the lens is able to act before the information loss happens, when you still have a million individual crisp images across the imaginary surface of the hole which can then be distorted before they get averaged together by the sensors. If you were to then take the blurred photograph and put it in front of the camera, you no longer have a million individual crisp images at each of these points of the hole, you have a million individual blurry images at each of these points of the hole, images of the photograph you took.

But! Here is one last fun and exciting idea. What if you take a million different blurry photographs of the same scene from different positions? Now you get into some interesting ideas that one might see, say, in astronomy or in tomography -- the process of trying to use a bunch of images of light or X-rays or what have you passing through a body, to reconstruct what's happening inside the body. And you might indeed be able to make several assumptions about a model to say "hey, the light hitting this sensor in this photo is a sum of the light emitted by these points, the light hitting that sensor in that photo is almost the same points except here around the edges, so if I subtract the one photo from the other I actually get a sort of more-precise view of the light emitted around the edges here, and maybe I can build up a 2D or 3D view that is less blurry than any of my particular images, by reasoning backward about what the original scene had to look like to make all of these different blurry photos at the same time.” So that sort of reconstruction is possible, and you can combine blurred photographs to try and unblur them, if you know that they were taken by identical cameras at nearly identical times and you know something about how they were positioned relative to each other.

Notes

1. So light is about a thousandth of the width of a hair, and atoms are about a thousandth the wavelength of light. Here’s the deal: most of what you know about light traveling in straight lines, actually is not what light naturally “wants” to do! Light at any point wants to go in all directions, so the light going down a hallway actually wants to bend around corners in that hallway. It is stopped from doing this by a phenomenon called destructive interference, where the light rays that do not take the shortest path are nearby other light rays which also do not take the shortest path, but they all pick up very different phases when they reach the target and so they all tend to destructively interfere, and only the light nearest the shortest-path (or a longest path!) all tends to have roughly the same phase and therefore constructively “stack up” into a substantial amount. This “rays of light” description is actually an interference phenomenon and if you try to make the hole too small, the light actually starts to do what it wants to do, to go in all directions, again.

2. Funny story, I discovered this actually as a child. When I was a kid, there was a time during which I was not aware that I needed glasses—nearsightedness kind of slowly grew on me and I did not realize that I was not able to see what others could see, that for others distant trees were not in fact green blobs but they could still make out individual leaves. But I still remember that in my classes I would put my thumb and forefinger of one finger up to my thumb of another finger and look through that tiny little pinhole. My teachers must have thought I was just being silly all the time, but that was not it: I could actually read better through the smaller pinhole I was forming!

3. So a camera actually has multiple lenses and controls the distance between them to have an adjustable focal length. It is true that one could adjust the screen's distance to also adjust the focal length, but in practice this is not how the eye focuses things. Instead the lens in your eyeball is connected to some muscles, called ciliary muscle, all around it that allows your eyeball to adjust the lens’s curvature by stretching it. The basic idea is that you have a bunch of strings on the edges of the lens holding it taut and flat for long-range viewing, and when you need to “accommodate” to see something close, these ciliary muscles contract the envelope—the not-lens thing that your lens is connected to by these strings. Doing so releases the tension in the strings and so allows the lens to return to a more round state, this is why it seems to be relaxing to look at far-off things and tense to look at things close-up. So your body does not adjust the retina, it has a mechanism to apply tension to release the lens. I assume that it is “backwards” this way because turning the strings into muscle fibers would require them to have an active blood supply which would make it really hard to keep the space transparent, but it might also be simply a historical accident, like the blind spots that you don’t know about because all vision is actually a hallucination.

Because it’s a 2-dimensional image.

Humans use binocular parallax (the slight difference in the image coming from both eyes) to perceive distance in 3 dimensions. The parts of the image that are convergent (i.e. the same in both eyes) are in focus, and the divergent parts (i.e. different in each eye) are out of focus. We change our focus by fine-tuning our pupillary distance to adjust the convergence — in other words, we cross our eyes slightly to focus on nearby objects, and relax them to focus on distant objects.

Since the entirety of the scene in a photo or video is the same physical distance from your eyes, parallax does not exist, and this fine-tuning is not possible.

In case you’re thinking about “3D” images, it’s not possible with those either, for the same reason — it’s still a 2 dimensional image that’s “faking” parallax to trick your brain.

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Just FYI, your understanding of how your eyes focus is not correct. The retina does not have any muscle tissue than can flex it. The only muscles in your eye are the extraocular muscles (7 muscles used for moving your eyeball), and iris dilator muscles (2 muscles) for adjusting the light aperture.

There’s also the macula which is an area of the retina responsible for detailed, high-resolution images, and the scotoma (the so-called “blind spot”) where you can’t see anything. Even if you’re focusing with just one eye, you’re still moving your entire eyeball slightly so that the image you’re trying to view is centered on your macula.

Most of the answers so far are inaccurate or wrong (I will explain why below). The answer is that it is not possible to deblur an image because photos and videos record only the amplitude of a light field and not the phase. A light field is characterized by amplitude and phase (and polarization, but that's unimportant here), but the phase is typically more important. This is dramatically illustrated by the following figure, taken from this paper:

Without getting too much into the details, what the author has done is passed two images through a lens and swapped the phases. The image formed from "dog amplitude + cat phase" (c) looks unmistakably like a cat, and the image formed from "cat amplitude + dog phase" (d) looks like a dog, which shows that it is the phase which stores most of the information in the light field here.

When you take a picture, you record only the amplitude. If your image is not in the focal plane, then the lost phase information is almost unrecoverable, and your eye cannot refocus the image.

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Note: I said above that the lost phase information is "almost" unrecoverable. There actually are techniques to recover this information, known as phase retrieval, but they are hard. This is an area of active research. The human eye and brain are not generally capable of doing this algorithmic task. Deblurring algorithms can be thought of as heuristic approaches to this problem which are adequate in special cases.

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A few comments about why other answers are incomplete or incorrect:

• Deblurring is not simply deconvolution. A blurry image can be thought of as a sharp image which has been convolved with a point spread function and then had its phase thrown away. Deconvolution is easy (and can be done by eyes via focusing and by brains in a variety of ways); phase retrieval is hard and generally can't be done by animals.
• This issue has nothing at all to do with binocular vision. The same considerations apply if you have one eye closed.
• This issue also has nothing to do with other sources of blur (e.g. aberration). The OP considers blur from defocusing, which is purely a matter of phase loss.

One other answer that is correct is that of @benrg, who approaches this question from the perspective of geometric optics. Phase information in wave optics corresponds to the direction of rays in geometric optics, and the fact that pictures do not record phase information is equivalent to saying that they do not record the direction of rays. The wave optics description that I've given in this answer is "more fundamental" than the geometric optics description, but they are both true, valid explanations of why you cannot refocus a blurry image.

For some "blurred fotos" eye can focus actually! Theese called "holograms", they preserve light wave phase, compared to usual photos, which stored only amplitude. Traveling light is complex thingie, only modulus of complex number is important for sensor as your eye, so you can't differ foto from live image, but argument of complex number is required to do refocus.

The answer to your question is the lack of information (lack of phase, definition per unit area) in the blurred image relative to a real (live image).

That being said, there is something called deconvolution, and there are two main types:

1. artificial, in this case a computer algorithm tries to make computerized educated guesses about what the original objects in the image looked like

In mathematics, deconvolution is an algorithm-based process used to enhance signals from recorded data. Where the recorded data can be modeled as a pure signal that is distorted by a filter (a process known as convolution), deconvolution can be used to restore the original signal.1 The concept of deconvolution is widely used in the techniques of signal processing and image processing.

https://en.wikipedia.org/wiki/Deconvolution

2. biological, in this case the human brain can recognize objects looking at a blurred image. Please note that this has nothing to do with our eye's focusing capabilities, but it is a capability of our brain processes

In certain cases the human brain can use information it stores (memories) to recognize blurred images.

On the image there are objects that are impossible to reconstruct using artificial deconvolution, but the human brain might recognize them.

Imagine you took a photo of a really dark scene, so dark the whole photo was just black. Could an animal with really big eyes look at the picture and see the original scene in full detail?

Well, we might not be able to ask the animal, but we can certainly try with photoshop, and no, you end up with a full gray image instead of a full black image, with lots of noise.

Why? Because the information you're trying to get out was never in the image. It was in the light (if you had big enough eyes or a big enough lens), but once the image is taken, that data is missing in the image made from that light.

Same idea with focusing. An unfocused image just doesn't have the detail needed to resolve the original scene in full detail. The blurry light that entered the camera had that information, but it was lost as soon as the photo was taken. If you'd had access to the original scene, though, and a strong enough lens, sure you could, but that's what eyes do :)