Radiant heat elements are basically a circuit, a closed loop, with electricity running thru them.
It works by resistance. The electrons are "squeezed" into a narrow flow, and the "friction" of the electrons is expressed as heat. Over time, if exposed to air, (and dependant on the material used) the material breaks down and loses some resistance and then they eventually fail, OR-
A flaw or damage to the "coil" will result in a bottleneck, creating too much resistance, and it burns itself in half.
That is a very basic explanation, but I think it describes it fairly well!