Why can we see light further than it shines?

The headlights that you see appear to have a brightness at your position on the road that is large enough for you to detect them. For the driver to detect you, he has to make use of the light reflected off of you back in his direction. That amount of light is small because you are not very reflective (as @Ankit alludes to in his comment) - your clothing absorbs and scatters a lot of the light incident on it.

Strictly speaking the following statement is wrong:

the car will brighten the road only few hundred meters further

The fact that you can see the light emitted by the car is evidence for the light illuminating all the way between you and the car. Two main reasons can be named why you see the car but the driver does not see you:

1. Scattering of light
2. The sensitivity of the human eye (and brain) for scattered light

The first effect leads to the car's light being scattered, the second to the driver not seeing this light.

You are seeing light that comes directly from the car headlights. In contrast, the driver is seeing light that comes from the car headlights and bounces off of you. First of all, you are pretty dull (physically, not personality-wise) and don't reflect a lot of light*. Second, the light has to travel twice as far to reach the driver's eyes**.

Edit:

*I think it's actually more important that you reflect light diffusely. If you were a mirror, and you were oriented so that you reflect light directly back to the car, then most of the light that reached you would reach the car. In contrast, since you're probably not a mirror, a lot of light is reflected in other directions, and only a bit of the light returns to the car.

**As commenters have mentioned, the intensity of light does not significantly fade with distance if the beam is narrow (as with a laser beam). This answer assumes that the car headlight does not produce a narrow beam of light, in which case the intensity of the light fades as $frac{1}{r^2}$.

Mainly because the driver sees the much brighter road right in front of the car, which is reflecting a greater portion of the light from the headlights. The light reflected from kilometers away is much less intense. The drivers eyes are not sensitized to the less intense light from farther away when they are mainly seeing the more intensely lit road in front of them. There is also the fact that the light you see from the headlights is much more intense than the light that the driver sees reflected off of you. This is due to doubled distance of the reflected light traveling with intensity lowering with the inverse square law https://en.wikipedia.org/wiki/Inverse-square_law#Light_and_other_electromagnetic_radiation , and because part of the light that reaches you is absorbed and part of it is not reflected directly back to the driver.

Well, we know that the density of light (flux density) decreases with $r^2$. We can assume that the car light is an isotropic source with power $P$. So we can see, that the flux $j$ at given distance, where there is a mirror is: $$ j_0=frac{P}{4pi r^2} $$ So, for understanding of the problem we can assume, that the light isotropically reflected from the mirror back to the driver. The flux density of reflected light, when it reaches the driver will be: $$ j=j_0 frac{1}{r^2}=frac{P}{4pi r^4} $$

The density of reflected flux depends on size, shape and albedo of mirror. But the main phenomena, why driver can't see the mirror, while the mirror can "see" the driver is, that the first flux decreases with $1/r^4$ and the other with $1/r^2$.

Let me try to combine the existing answers into a full picture.

There are four effects which make you perceive light reflected from a distant object much weaker than a direct light at the same distance:

1. Only a small fraction of light is reflected from most objects.
2. The light is reflected isotropically, so only small fraction of light is reflected in the direction of the car.
3. The total distance traveled by light is twice the distance between the car and the object.
4. Pupils adjust the vision sensitivity to the brightest object in your view: that will be the objects close to your car. This makes farther objects harder to see, even if they are somewhat lit. If you in turn look at a distant car, the brightest objects will be its lights and you will see them easily.

(@Vid provides calculations explaining points 2. and 3.)

This is why retroreflectors (like street signs, or reflective vestments) are so much more visible, because they address points 1 and 2: they reflect the light well, and they reflect it at the direction of its source. At the same time presence of a street sign, or incoming car lights in your view may make other objects even more difficult to see.

There are two main reasons, as alluded to throughout the other answers:

1. Your eyes and the drivers eyes being open different amounts--due to having to calibrate with the amount of light in the environment in order to not over or underexpose the environment.
2. The amount of light coming from you vs. coming from the car.

In the case of 1, look at your pupils when in a very dark room, vs very bright room. You'll notice that in the dark room, they're much larger. Why is that? Well, there are less photons coming into contact with your eyes when its darker, so in order to maximize the light coming into your eyes, they open up more. In your scenario, the person standing on the dark road will have larger pupils than the person in the car, because they're calibrated to the dark environment with little light.

In the case of 2, you're less bright than the car. Even if your pupils and the drivers pupils were collecting the same amount of light, there's much more light coming from the car (which is producing light), than you (which are only visible due to reflected light).

I have never seen such a load of unnecessary verbage in answer to a question! We see the light from stars millions of light years away from us but only a complete dum-dum would expect an observer on a planet orbiting a distant star to see that star's light reflected off say an astronaut doing an EVA in Earth orbit. The answer is that there is simply not enough light reflecting back towards the source. It's as simple as that.