Why does FM sound better than AM?

What I’ve never understood is how the FM receiver can lock on to the signal if its frequency is always changing. Doesn’t the receiver need to lock on to something? If the answer is “it locks on to the amplitude, which doesn’t change,” well AM is bad because the amplitude is subject to interference, so wouldn’t FM have the same problem?

One possibility is a phase-locked loop. I don't know if there's anything better. It matches the frequency of a voltage controlled oscillator to the frequency of the incoming signal by detecting the phase mismatch. Then, the control voltage for the VCO becomes the audio signal.

I don't think radios use a PLL to demodulate the FM audio: the signal has a huge "pilot" tone at 19kHz that you can match to get the first part of the spectrum, mono audio (L+R channels), and at double that frequency you know you'll find the stereo part (L-R channels). Precise phase estimation is only necessary to decode the RDS digital data (station name, datetime, etc.).

They do.

First of all, the pilot is only required for decoding stereo and RDS. Mono FM does not use a pilot, so obviously there had to be a way to detect FM before stereo came along. I linked to a few of the approaches in a sibling (cousin?) comment.

Second, the pilot is embedded in the decoded FM audio. You need to demodulate FM to get to it in the first place. If you look at the waterfall display in an SDR receiver, it might seem like the signal is already present in the original radio frequencies (especially during silent periods), but it's there only indirectly.

If you have silence in an FM transmission (say 96.6 MHz), the only audio component present is the 19 kHz pilot signal, which causes the FM radio signal frequency to vary between 96.6 MHz ± k*19 kHz (not sure what's the value for k, but it's not 1). The sine likes to spend most of the time near the extreme values of its range; plot a histogram of a sine wave and you'll see peaks on either end.

The waterfall is basically a histogram over frequencies so it gets those peaks as streaks on both sides of the main carrier frequency (plus smaller ones for other components in the signal).