What do the base ingredients in pancakes do?

The range of possibilities with a basic pancake recipe is not very broad. There are additional things you can add of course, but the basic recipe is really basic.

Eggs are mostly protein. When you cook an egg, it goes from liquid (basically) to solid. The egg blended into the pancake batter will do the same. Pancakes are generally cooked pretty quickly over a fairly hot fire, and that quick heating makes the egg proteins lock up and stiffen. Think of the difference between making a fried egg and scrambled eggs: fried eggs should cook fast, while scrambled eggs cook slowly to keep the eggs from being rubbery. (The other pancake ingredients counter that rubbery toughness of fast-cooked eggs.)

Flour is mostly starch. Starch absorbs moisture (from the egg and the liquid). People started making things like pancakes because flour is relatively cheap and keeps well, and one egg can make a lot of flour-based pancakes and feed a lot more people than the egg alone.

The liquid (milk in your recipe, but other stuff works) is necessary to thin out the batter; we're still stretching that egg out. Without a liquid addition, you'd have a lump of thick dough. By keeping the pancake batter thin, you ensure that the batter poured onto the hot griddle will remain loose enough to spread out into the familiar flat disc, providing a large surface area for the egg protein to solidify.

The salt is there so that the pancake doesn't taste bland. That's an easy one to test. Most American pancake recipes also include some sugar or another sweetener.

American pancakes without some additional fat — melted butter or vegetable oil, usually — would not be liked by most Americans. The added fat makes the pancakes softer and richer. Using table cream or something instead of plain milk can work too, as would adding an extra egg yolk.

Baking powder is a mix of dry sodium bicarbonate and one or more acidic salts. The blend is (usually) designed so that some amount of gas is produced as soon as the powder is mixed with liquid, and more produced when the batter is heated.

Finally, besides the bill of ingredients, the process used to combine the ingredients is also important. Wheat flour has gluten in it, and too much mixing in liquid will form gigantic protein mega-molecules in your batter and result in disturbingly chewy pancakes. You want gluten development for bread, but not pancakes, muffins, and other "fluffy" baked goods.

Generally, dry ingredients (except granulated sugar) are pre-mixed separately from wet ingredients. The two are combined as the last step before cooking.

You also want some air in the pancakes (for American pancakes at least), and that's of course what the baking powder does. You can get even more air in by separating the eggs, using the yolks as a "wet" ingredient, and then whipping the egg whites into a soft foam. The foamed whites are then blended in when the wet and dry ingredients are combined. (Note that you don't want air in crepes, and that's the purpose of letting the batter sit for a few hours before cooking: it lets bubbles escape, bubbles formed in the blender or via the whisk used to prepare the batter.)

Milk also helps the pancakes brown. Pancakes made without milk and egg are usually very pale.