Here is what happens, chemically, when you bake a pizza. (Warning: The following description will be ever-so-slightly disgusting.)
The water molecules contained in the cheese atop the pizza—all those little H2Os, trapped between the protein and the fat—heat up. In fairly short order, the water begins to boil. When that happens, the water becomes steam. But the steam is trapped in the cheese—inside all that protein and fat—so it can't evaporate into the surrounding air. Instead, it pushes against the surface of the cheese. The cheese, in turn, starts to bubble.
So that's what generally happens. What that basic process looks like in practice, though, varies greatly depending on which kind of cheese tops the pizza. If the cheese has more moisture, the bubbles the steam creates will be large. A less elastic cheese will produce smaller bubbles.
And that distinction, in turn, affects what is arguably the best part of a pizza: the cheese's ability to brown in the oven. All that bubbling and steaming causes the oil to leak out of the melting cheese, settling on the surface. Cheese with high moisture content and low fat content will create bubbles large enough to break that surface of oil, exposing the moisture in the bubble directly to the oven's heat—meaning that it evaporates, leaving the rest of the cheese to brown. Only some cheeses produce that effect, though. If a cheese is low-moisture and low-fat, it will burn; if it's high-moisture and high-fat it will simply stay greasy without browning.