What cause the Garbage disposal wire damage showed in the picture?
That does look like heat damage. It's possible that if the disposal jammed and the cord was pressed against something that the current draw heated the wiring enough to melt the housing like that.
UNPLUG THE DISPOSAL AND DON'T USE IT UNTIL THE CORD HAS BEEN REPLACED
Under your sink, in a wet location, having exposed wires is very dangerous. Your disposal may have a replaceable cord-set, which means that you can unscrew this old one from the disposal and install a new one by screwing it in, or, if it doesn't you might get away with cutting off the damaged piece and installing a replacement plug. Worst case, you may have to replace the whole disposal.