From what I remember, it's because the motor never gets started right.

V = I/r

When the voltage is low, the motor can not turn over at all (doesn't start), or will run more slowly than it should. If there's no actual movement, then there's no load on the coils of the armature. If movement is slow, then there's very little load on the coils of the armature. Voltage doesn't change, but resistance goes down, so current spikes. High current = high heat, and the motor burns out.

I don't have a firm grasp of the electromagnetic principles that say why no/slow movement of the motor assembly results in low resistance, sorry. I know when we were in Jamaica, our fridge was plugged into a box that would cut out if the house voltage was low. Figured it was better for the fridge to be unpowered for awhile rather than having to keep replacing the electric motor.