Why do we use eggs in cake 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.
Hobodave's edited title is more appropriate, I think, as many delicious cakes do not have any eggs. Eggs have different effects on different cakes, but I would argue that if your dish is simply "cake," then they aren't, strictly speaking, necessary. It's often possible to replace eggs with other ingredients to achieve the same desired effect: adding more baking powder, protein and fat is usually a good place to start.
However, it's absurd to talk about what role eggs play in "cake," because this has different answers for different cakes. Eggs in a pound cake play a completely different role than eggs in angel food cake, for example.
Coagulated egg proteins in combination with gluten give baked goods the supporting structure they need.