Why does FM sound better than AM?

Someone gave me an analogy some time ago that made a lot of sense.

If you shine a flashlight through a tree blowing in the wind and vary the brightness to convey information, the signal can get distorted pretty easily.

However, if you have a constant brightness source and vary the color, it’s a lot easier to figure out what the source is trying to convey.

I always shy away from analogies because more often than not they give the wrong "feel" for a concept. But this is one of those rare exceptions.

It's not an analogy. This is precisely how it works.

Unless your car radio consists of a flashlight and a tree, this is an analogy.

Essentially that is actually the case. Human-visible light and AM/FM radio waves are just different wavelengths along the EM spectrum.

A flashlight beams out waves that we can see; a radio transmitter beams out waves we can't. The brightness of the beam of light is related to its amplitude, just like the signal content in AM radio is related to its amplitude. And the color of the beam of light is related to its frequency, just like the signal content in FM radio is related to its frequency.