Why does a thermostat turn a house's heating unit on and off?

The short answer is that it's what it was designed to do. It is so designed in order to keep an indoor area at a constant, comfortable temperature.

Inside of the thermostat is a coil. The coil expands and contracts with the temperature. When the coil expands to the point it's supposed to, it makes contact with an element detached from the coil, thus allowing an uninterrupted circuit to be made, which in allows power to get to the HVAC system.