What role do eggs play in baking?
There are basically two types of cakes: butter and sponge (or Génoise).   In a butter cake the egg proteins, like the gluten proteins, help build the structure of the cake.  Additionally, the egg yolks have emulsifying action, helping create a smoother batter and more stable air bubbles.  In a sponge cake the eggs additionally act as the main leavening agent, helping create and retain the bubbles that will expand during baking.

The eggs also add fat, flavor, and color to the cake.