Emulsification is sometimes a word that gets thrown out as a general functionality. The basic principle of emulsification is modifying the surface tension between water and oil. When combined, this is then an emulsion. Like a creamy salad dressing, mayonnaise, hollandaise sauce, etc. The modifying of the surface tension also allows for the formation of bubbles to capture gases. Think about blowing soap bubbles. Soap is a surfactant, which is the short name for surface active agent.
Eggs contain protein in the egg white, to provide emulsification and aeration, and the egg yolk has lecithin, which provides emulsification to mix fat and water to a creamy consistency. Eggs provide fluidity for mixing and batter flow, aeration from mixing, expansion during the baking process, and structure for the baked product when the eggs coagulates and cook. There is a lot of complexity in replacing eggs for various baked goods. Cookies tend to be among the easiest for functionality. Replacing the nutritional protein is often part of the challenge in formulating today as well.
Hope this is helpful.