A recording of a sound wave is a pattern of samples. Those samples have positive and negative values, which correspond to the compression and rarefaction of the source sound.
As I understand it, "reversing the phase" means inverting these. Each sample is its same magnitude, but if it was positive it is now negative, and vice versa.
This is why reversing one channel of a stereo sound causes cancellation - there's generally a lot of overlap between the two channels, so the negation of one causes them to appear canceled when heard together.
It's worth mentioning that in most stereo recordings, the two channels aren't exactly the same, so it's not 100% cancellation. But there's usually a fair bit in common.