Power is the product of voltage and current. Less voltage means more current to produce that level of power. Conductors are generally sized for the designed level of current, so they overheat when asked to carry more.
When operating at the proper voltage and current, the right amount of torque is produced and proper speed is maintained. When voltage is reduced, torque is reduced and therefore the amount of slip in the motor is increased resulting in lower speed. Since the speed is reduced, back EMF is also reduced causing higher current than normal. It is like trying to run a 4 cyl gasoline engine on 3 cyls. It is just harder on the engine.
According to what I've heard, at least for certain kinds of machines, the lower voltage results in the slower speed, and that means less cooling by air of moving coils, which can then overheat.