Feedback is actually a quite simple thing. As the name suggests, part of the output signal is "fed back" to the input of the poweramp. This signal adds to the signal coming from the preamp and is amplified by the poweramp again. The poweramp now has more overall gain.
"Negative" feedback means that a phase-inverted version of the output signal is fed back, that is, everytime the input signal is positive, the output signal is negative and vice versa. In effect this does not add to but subtract from the input signal. In other words, the overall amplification of the poweramp is decreased (hence the word "damping").
Why would some amp designer want to do that? The answer is that negative feedback also decreases nonlinearity in the poweramp. Consider what happens when the amp starts to distort. That is, the power supply isn't able to deliver the voltage which is necessary to linearly amplify a very high input voltage. The actual output is a bit lower -- and so is the signal in the feedback loop! There is less signal subtracted from the input, the overall gain is slightly higher for that input voltage. The feedback loop "linearizes" the poweramp a bit. That's why the amp sounds cleaner and tighter when you increase negative feedback.
How negative feedback affects harmonic distortion
Reducing negative feedback increases harmonic distortion including 2nd and 3rd if operating below clipping. This is because feedback linearizes the amplifier. The Axe-Fx has a non-linear "power amp" just like a real tube amp and if you turn down the feedback (damping) then the power amp will be less linear and will therefore introduce some distortion products. The distortion products tend to be low order (2nd and 3rd).
Once you enter clipping, negative feedback can actually increase the distortion products since the onset of clipping is sharper. A power amp with no negative feedback (i.e. AC30) will produce distortion even when not clipping. However, it will go into clipping gently since the transfer function tends to be smooth. A power amp with negative feedback will produce less distortion when not clipping but will enter clipping more abruptly since the smooth curve is essentially turned into a piecewise linear function with an abrupt gain reduction at the supply rail.