Hi,
This is all about how waves add together, usually just two sine waves.
The reason for all the concern is because the feedback goes right back to the input, so it adds to the input. The feedback comes from the output, so if you have the output go back to the input and it adds up to more than what went in the first time, then you get a higher output, and if then THAT new output goes back to the input, it adds up to even MORE output, so the output keeps growing and growing until the output starts to reach the power supply level and then you either get real bad distortion or you get an output that remains high or low, so the output becomes useless for most applications when this happens.
Now if it was DC, DC just adds to the input in a given ratio, but since we are concerned with frequencies there are two sine waves: the input and the output coming back to add to the input as feedback. If the two sine waves add up to a level that is too high, the output goes into saturation and locks up or else produces a huge amount of distortion.
Sine waves are a little more tricky than DC because they have a phase associated with them. So we have to think about what happens when two sine waves add together. The phase now has a lot to do with it as well as the gain, because two sine waves that are in phase add just like DC, but two sine waves that are out of phase by 90 degrees do not add the same way. For example, if we had a DC level of 1v added to another DC level of 1v we would get 2v. If we had two sine waves with a peak of 1v and they were in phase we would get 2v peak output, but if they were 90 degrees out of phase we'd get a peak again of 1v, just with a shifted phase. So in the 90 degree case we dont see the amplitude increase, while in the 0 degree case we see it increase by two times the value of one sine peak, and that is what could cause oscillation which could lead to saturation. That's basically why there has to be a phase criterion as well as amplitude criterion.