How do I explain that post-processing is an important step in producing good photos?

One thing you might try is asking them to show you an example of a great photo that hasn't been "'shopped". If you define it as any photo that wasn't exactly as it appeared when the shutter was actuated, they probably can't. Just as the decisions made in the darkroom had a great effect upon the finished product in the film era, the decisions made at the computer monitor in the digital era do the same thing.

Art critics have called Ansel Adams' photograph of "Moonrise, Hernandez, New Mexico" one of the greatest photos ever. Art historian H. W. Janson called this photo "a perfect marriage of straight and pure photography". Yet if you examine some of the over 900 prints that Adams produced from the negative himself it is very clear that over the course of more than two decades he produced a series with a remarkable range of variation. Adams explored the relationship between the various elements in the scene until he finally seemed to have found what he was looking for. The prints we now view as "definitive" did not appear until the 1960's. Adams snapped the image in 1941.

If your friends are not aware of the history of photographic technology there probably isn't a whole lot you can say that will change their minds. A picture that has been "photoshopped" is viewed as less than legitimate by the uninformed masses. Many people today will view a photo and instantly label it as "'shopped" so they can refuse to accept whatever statement the photo is making, deny the historical event it depicts, or reject the premise that such a photo could be produced without "photoshopping" it if they can't understand how it was taken.

In the end the only person you must satisfy as an artist is yourself. Did the final product realize the vision you had when you took the shot?

Welcome to the lonely world of the "Artist"!

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Artnet.com article about the history and current market values of "Moonrise, Hernandez, New Mexico"

Wikipedia article for "Moonrise, Hernandez, New Mexico"

A photo of Adams late in life posing in front of two prints of "Moonrise..."

You might have walked into the trap that I and most others have: we overdo it!

Because we have seen the dull, almost monochrome, raw image, developed from default settings, we really feel the need to crank up the "power" (saturation, contrast), to make it vivid like those film posters and arty photos we see all over the place.

A good start is to crank it up and then take it back 50%. And if you are colourblind like me, take it back 75% :)

Although an answer has already been accepted, I will allow myself to add one more unsaid argument.

Post-processing is a very important step, especially in the field of digital photography. The point of post-processing is to a) correct photographer's errors (horizon line, exposition), b) enrich a photograph with specific mood by altering its color gamut or introducing cropping, c) make it look natural. At least the last point requires some explanation.

A digital camera has a different working principle than a human eye. It captures light linearly (light is captured by a counter by every pixel on the sensor) whereas human eye sees more detail in light areas and less so in darkness. We say that a human eye has a non-linear color gamut. This is why mid-day shots appear so burnt-out when shot with a digital camera. White balance is another similar story where camera has to apply algorithms to deduce the correct white balance. Human eye adapts automatically. One could say that post-processing is a built-in option in human brain.

To make a photograph look "natural", we alter the light curve, change white balance, enrich saturation and play with a number of other settings. If this is done qualitatively enough, a person unfamiliar with photography will not notice anything. For him/her this will be a standard, maybe "good-looking", photo.

Post-processing is fully justified in cases when it remains unseen (like in the photo by Ansel Adams quoted in the answer by Michael Clark) or when it adds artistic effects. When neither of these applies, when people can not see artistic in the photograph, they tend to seek for the natural. I am afraid that when you hear people tell you your photo has been photoshopped it might indicate that you have altered it far too much so that it looks neither artistic nor natural.