many ways, here's what ppl run into the most, a voltage spike, where the voltage unexpectedly increases dramatically over a small time.
high voltage makes the insulation useless, effectively speaking, the "capacitor" model of 2 wires next to each other comes into play. at the moment of a voltage spike, there is a discharge across 2 wires. This discharge actually causes a current spike. The current increases the temperature of the location of the spike causing burns and burn marks. It's like a small explosion. a wire can also be a trace on a board. typically we see voltage spikes causing current spike.
V = IR, that is, the voltage is the current times the current. if you take both sides to change over time, you get change in voltage over time is the change in current over time multiplied by the resistance which is constant over time. the voltage spike is a high change in voltage in time, resulting in a high current change over time, which is a current spike. typically this is what a power surge is, where P = IV, power is the current times the voltage. (all this crap is in magnitude).
the amount of overvoltage depends on the design and materials used in the circuit. the voltage spike is also why circuits get damaged by static electricity. The discharge, when you touch something, is in the order of a few hundred to a few thousand volts over a very small period of time (electrons travel at the speed of light).
why am i talking about current? because for a fixed wire thickness, the more current you run across it, the HOTTER it gets temperature wise. Thicker wires have more mass to dissipate the heat. this is why cables are rated for current (amps) and not voltage. the insulation is rated for the voltage. also, as the temperature of most insulation rises, it's ability to insulate decreases. so, basically this is a lose lose proposition. your current increases, and the temperature of the wire increases. since the insulation touches the wire, the temperature of the insulation increases. The ability of the insulation to insolate (prevent shorts) decreases. So, after a while, the insulation breaks down, and a short happens which causes a power surge through the point of insulation failure, and the insulation catches on fire, as well as anything near the point of insulation failure. this is how most electrical fires happen. the wires get too hot, and the insulation becomes useless. the damage is over time, since insulation doesnt really repair itself. it kinda melts a little if the wire gets overheated repeatedly.