Phase is one of those things that is a lot more complex in reality than it first seems from the theory. Sound waves are transmitted through oscillating air pressure. Electronically, it is captured as a signal that maps the pressure changes to changes in electrical voltage. If you had a fixed frequency wave, it would look like a sine wave going up and down in regular areas of positive and negative pressure. You can even see this when you look at a large speaker and see the cone moving in and out.
So what happens when two audio signals meet? Keeping it simple with a single frequency, it is easy to see that it depends on how the signals are aligned. On the wire, if you have a signal that is at +1 volt and a signal that is at -1 volt and you combine them, you end up with 0 volts. If you instead had two signals that were perfectly lined up, then the 1 volts would be added to each other and you get a much higher amplitude wave.
It could also be partially aligned which results in a much more complex waveform than the original sine wave that has bumps in it that make a frequency other than the original frequency of the wave.
When listening to a similar situation from two stereo speakers, it gets even more fuzzy because the sound arrives at different times at both ears and the alignment at each ear differs slightly. The spacing of the speakers and the listener also matter for this. The result can be drop offs in SPL at certain spots in a room.
Phase is a measure of the difference in timing between the two signals. Ideally, 180 degrees out of phase means that the wave is an inverse of the other wave. A phase inversion simply flips the signal to try and get it closer in line with another signal.
The reason that each channel has it's own phase adjustment is that it allows either a slight delay, or a reverse of the - and + sides of a signal, in order to make two signals be more in-phase. This is normally done to correct for the problems I described earlier.
The thing that makes it even trickier is that we don't deal with single frequency sine waves. We deal with complex waveforms that have multiple frequencies in constantly varying intensities. Two different samples don't match up exactly and so phase becomes a much more subjective thing to listen for and adjust to minimize the occurrence of phase related problems.
There is analysis methods that help determine if something is out of phase, but in the real world, it's still a rather complex signal analysis task that is often best handled by learning what out of phase sounds like so that you know when to adjust it to minimize problems.