An AM receiver is a machine that senses the amplitude at a specific em frequency. In this situation, noise and interference become random additions or subtractions to that amplitude. Draw a sine wave, then go over the line with vertical ticks or scribbles. Now imagine taking a random sampling of points and reconstructing the original wave perfectly (without a computer). Most of the information is just gone and you end up with a noisy output wave.
Now an FM receiver is one that measures frequency changes above and below a 'carrier' frequency. The amount of deviation away from center represents the amplitude of the sound signal being transmitted. In this setup, noise and interference are also random additions to the amplitude, but also at random frequencies. On average, interference happens evenly over the entire range of frequencies you're looking at. That means that the highest amplitude is still the same frequency away from center, it just has a slightly different amplitude.
Go back to that sine wave. You can't see the original signal behind all the noise, but you can still see how far apart the peaks are. You can still easily extract its frequency content.
FM uses the frequency dimension to transmit data because random noise can't really affect frequency. Noise mostly happens in the amplitude dimension across all frequencies at the same time.
FM is more robust because it uses two dimensions to encode information vs AM's single dimension. That's also why FM is in stereo!