TL;DR: use one tablespoon of flaxseed meal per egg as a drop in substitute for eggs in baking. To avoid clumping, soak the flaxseed meal in an equal volume of water first or add the flaxseed meal to the recipe last.
Eggs serve a few main purposes in baking, they:
- provide structure
- provide some leavening
- provide water, macronutrients, and flavor
- function as an emulsifier.
When the proteins in eggs are heated, they denature and bond to each other, and form a network. This is how eggs give baked goods structure, their protein network. Other food products that provide structure by forming polymer networks include gluten, pectin, and something in flaxseed (unfortunately, I'm not sure what exactly). Of those three, flaxseed gives a structure most similar to that of eggs, but it needs to cool back down before the structure fully forms.
When eggs are whipped, they form small bubble of air. When these bubbles are heated, they expand, which causes some expansion. The amount of leavening caused by eggs in most baking recipes however, is insignificant.
Eggs provide some liquid to recipes, which affects the consistency of the batter a little. Additionally, they provide some fats, which hinders gluten formation, and stops baked goods from becoming tough. In recipes without a source of fat besides eggs, some oil may need to be added.
Finally, eggs act as an emulsifier, which means they prevent mixtures of water and oil from separating. In most baked goods recipes, this doesn't matter, because batter will not separate anyway.
Because baking recipes usually rely on the structure provided by eggs much more than any of the other properties, flaxseed meal usually makes a good substitute for eggs in baking.