Can anyone explain how this circuit works?
It's a very simple thermostat. One of those pieces of metal is actually bimetallic. The dial controls the distance between the top piece of metal (call it a "setting level") and its stop. The bimetallic element will actually curl away from the top piece as it heats up. If the dial is turned to a low setting (far away from the bimetallic element) then the bimetallic element just has to curl a little to "get away" from the setting level, thus breaking the circuit and stopping the heating. If the dial is set up high, then the bimetallic metal has to curl a fair amount to get away and break the connection.

As the bimetallic element cools, it straightens back out and will eventually make contact again, starting the heating process over.

Those little round discs are the contacts. Sometimes they are mildly magnetic in order to help "make" a good solid contact to start the heating process.