the eternally hot core flashes water to steam to drive the turbines which drive the generators which drive the electricity.
Most nuclear power stations use pressurized water reactors. The primary coolant is water that is prevented from vaporising by keeping it under tremendous pressure. As it passes through the reactor its temperature rises by only a few degrees. It then passes through a steam generator where it heats unpressurized water into steam. This is relatively low grade steam and the turbines are specially designed for it.
As a matter of fact, if the water in the primary coolant loop starts getting vapor in it, it becomes suddenly very inefficient at absorbing heat from the reactor, and you've got a big problem.