Thermal contact conductance is all about how hot things flow. It's sort of like how water moves in a pipe, it gets hotter if more water goes through. Thermal contact conductance is like that, but for heat instead of water – it is the rate at which heat moves from one object to another when they are touching. A good example of this is a hot pan on the stove. Heat from the fire beneath the cooking surface will flow up through the pan and make it hot. The better the thermal contact conductance of the pan and stove, the faster the heat will move and the hotter the pan will get.