Answer
AC power reverses the current repeatedly at a high enough speed that there is not enough time in between for the light to go out.
Work Step by Step
The voltage goes up and down in a sine wave fashion, and every time the voltage direction reverses, the potential energy of the electrons is raised. In other words, with every trip through the light bulb filament, the electrons give up some potential energy to the filament (which outputs heat and light), but get it back again.
