A less technical explanation is that the opamp can "hear too much of its output" and follow it faster than it is actually able to.
An opamp is basically just a differential amplifier with very high gain. It is taking the difference between the + and - inputs and amplifying that onto the output. This gain is really high like 1000000x so that you can add external resistors for feedback and bring the gain down to whatever value you want (if the gain was small you would have to factor it into your feedback calculations). This also stabilizes the exact value of the gain to exactly what the feedback resistors set it for.
Feedback indeed makes PA speakers sequel into a unstable oscillation when the microphone comes too close. This is an example of positive feedback. It causes a signal to around and around getting amplified larger and larger every time. Negative feedback does the opposite, makes it smaller, hence any oscillations die out. The difference between the two is phase. Negative feedback is 180 degrees out of phase.
The stronger of a negative feedback you give the more the gain will be suppressed, all the way down to 1x unity gain. This means the opamp is taking a larger proportion of its output as input and looping it around. So any output disturbance is quickly countered by outputting the opposite of the disturbance. However some opamps might not have an output fast enough to counter that disturbance as quickly as the opamps input stage is commanding it to do. The reason for the output being too slow might be because the opamp is just designed this way (they can use the extra input speed to make the opamp faster at higher gains) or because you are loading down the output too heavily (like too much capacitance on the output).
So once you have this case of the output stage not keeping up, you get the input stage not seeing the correction to the disturbance being applied so it commands a even larger correction, then once the output stage catches up and outputs said correction, the correction is too big, so the input stage commands a negative correction, but it is not seeing the result yet, so it command an even bigger negative correction, overshoots the other way, then back up...etc So the amplifier starts chasing its own tail, resulting in a oscillator. This slight output delay is what causes a phase shift, causing the negative feedback to not be 180 anymore, making it look more like positive feedback.
By setting the feedback for more gain, you are attenuating this negative feedback, so the input stage sees less disturbance from the output, so it commands less reaction to the disturbance, so the output has a easier time keeping up to what it is commanded to do.
Hope this less technical explanation is more digestible than the more math-y explanations that are usually out there on this.