Why does FM sound better than AM?

That's not the real story. The RF environment is noisy, with naturally occuring static "sparks", but also with manmade RF noise.

This static and RF noise is AM. It's impossible to filter it out from an AM signal, and so the background noise gets amplified with the signal.

Encoding the signal in a modulated frequency (FM) means we don't need to amplify the detected AM signal and it's associated background noise.

That's exactly what the parent comment described in the beautiful example where the AM noise is due to moving tree leafs affecting the intensity of transmitted light, and you can fix it by varying color, which means varying the frequency spectrum of the light.

It's not that simple, though. The only way you can detect frequency is by measuring the amplitude (and then differentiate; except of course in an analog circuit, you don't do that exactly, you have some mechanism that tries to track the carrier wave smoothly instead), so amplitude noise will necessarily also become frequency noise. But generally white AM noise will be pushed upwards in the spectrum after FM demodulation, away from the area where you care. (You can also add a hard limiter, which amplifies this effect; even more noise high up, even less noise further down.)

How does the radio follow the frequency modulations if the radio cannot "see" at a specific direction?

In the example, the amplitude of the flashlight signal is distorted by the movement of the trees. The signal is never completely hidden. Not sure if that answers your question...