Generally, when we bake something like a cake or a loaf of bread, we need the mixture to stick together as a chunk AND to rise. To help the mixture stick together as a single chunk, we need some kind of liquid to mix with the flour so that proteins (very small particles) in the flour can form crisscrossing "nets", and milk can be used as this liquid (we can also use water or other types of liquids).

There are also proteins in eggs, so eggs can also help hold the cake/bread together and help the mixture turn into a solid. Additionally, eggs have a type of fat that help other fats, such as butter/shortening/olive oil, mix with water-based liquids, such as milk.

Without proper mixing, the baked goods would come out uneven or uncooked in certain places. That being said, we can bake things without eggs or milk if we just need the flour to stick together; we can bake things with just water and flour, which will stick together and even rise if we give time for the natural yeast in flour to work, but the baked product of just flour and water would not be what we think of as cupcakes.

You could probably use soy milk to make cupcakes, or maybe even water. Eggs are good for making the other ingredients stick together in the cupcakes.

There are recipes for making cupcakes without eggs and milk. This one doesn't have butter either: cupcakes recipe without eggs and milk.

Cake includes fats and proteins to keep its moisture and texture, which eggs and milk have but the other things that go into cake doesn't. Without eggs and milk, you would turn your flour into bread instead of cake.

Two major roles of eggs for baking cupcakes are that of binder and that of leavening agent.

Being a binder essentially means that eggs are what hold the cupcake together. Removing the eggs means that there is less holding together the rest of the ingredients. One can see how this would be problematic in trying to make cupcakes.

Leavening agents add volume. Without a leavener, the cupcakes would be smaller, flatter, and denser (though cupcake recipes often include other leaveners besides eggs). Other contributions of eggs are tenderizing the crumb and adding moisture.

Like eggs, milk makes several contributions in baking.

One of the more important properties of milk in baking is that it is wet. This allows it to hydrate the dry ingredients so that chemical reactions can take place and dissolve the sugar (which helps to distribute it). This is not exclusive to milk though; water will also hydrate the dry ingredients. However, milk also contains proteins, fats, and sugars. The proteins help to strengthen the batter so it can better capture gas bubbles which make it rise, and also contribute moisture. The fats and sugar help with absorption of water and lead to a moister, more tender product. They also add flavor, which water lacks.

Further information on the roles many baking ingredients can be found on ScienceLine here.