What do eggs do in enriching and glazing
Glazing has something to do with browning (a.k.a the Maillard reaction) and also the fats and perhaps other substances in the yolk. 

When you bake a pastry covered with eggwash, the high heat cooks the egg to an extent that changes the chemical properties of the sugars and proteins in it. The result is golden-brown, lightly shiny and quite falvorful coating. The chemistry behind it is the same one that browns steaks and toasts in high-heat. I'm not sure about the fat's role in the glazing, but eggwash made from whole eggs or just the "fatfull) yolk produce shinyer glazes than what the low-fat eggwhite produces (but it still does!)

As for "enriching": as rumstcho commented,"enriching" is a loose name for many different things. In mayonnaise, for example (a vinigrette sauce with up to 10 times the amount of oil, "enriched" but to be more percise emulsified with eggyolk), the egg just gives its lecitin - an organic chemical that binds oil and water. It's thanks to the lecitin that mayonnaise is so rich. Without it you'd have a glass of oil with a cup of water in it.

But in other cases the "enriching" is indeed made by coagulation. I'll take another example from the world of sauces: when you make hollandaise sauce, the creamy texture is achieved by beating the eggs while you lightly heat them, and thus partially coagulating the proteins.

Eggs are a true magic. I'd mention that it's one of nature's few original, dedicated foods, but the last time I wrote that I was oddly trolled. But do ask further.