Chemically speaking, what happens when you temper an egg? Why doesn't it simply cause the eggs to solidify or scramble?
One of the reasons is what cantido probably meant: you can overheat the eggs quickly if you pour them into the hot milk. Heat transfer is proportional to the distance from the boundary between the two materials. If you pour a thin stream of hot milk into tepid eggs, most of the eggs don't get heated, because they are far away from the boundary. They only come into contact with the milk after it has cooled down a bit. Also, the milk you pour is somewhat cooled down before it hits the eggs because 1) there is no burner underneath it (which may still be on, or, if you are not on gas, will give off considerable heat even after turning off) and 2) it cools down while being poured through the air. Thus tempering prevents the localized overheating which would happen if you would add small amounts of egg to hot milk.
But there is also some magic that I can't explain, although I've seen it referenced in reliable books (Cookwise). The temperature at which eggs overcoagulate is dependent on the speed with which they are heated. The coagulation of an egg is a slow process, and there are many intermediate stages along it, from absolutely liquid to absolute rubbery. But the weird thing is that this process's speed depends on the rate of heating - eggs which have been heated from 25 Celsius to 80 Celsius slowly can create a pleasant soft custard, while eggs which have been heated from 4 Celsius to 80 Celsius quickly can curdle the mixture. I wish I knew why this is so, and hope that somebody else will provide this final piece of the puzzle in another answer.
So, by tempering, you achieve a slower, more even heating, which extends the temperature interval between raw and curdled, and gives you a pleasant texture overall.
As for the chemical process of thickening-and-curdling in eggs, Chef Code explains it well. It starts with a soft web of coagulated protein which loosely holds the liquid trapped (tasty custard) and ends up with a very strong web of very deformed protein which expels liquid (curdling).
It does not provide the chemical composition changes you might see in a lab although. I hope this is helpful.
Egg proteins change when you heat them, beat them, or mix them with other ingredients. Understanding these changes can help you understand the roles that eggs play in cooking.
Proteins are made of long chains of amino acids. The proteins in an egg white are globular proteins, which means that the long protein molecule is twisted and folded and curled up into a more or less spherical shape. A variety of weak chemical bonds keep the protein curled up tight as it drifts placidly in the water that surrounds it.
When you apply heat, you agitate those placidly drifting egg-white proteins, bouncing them around. They slam into the surrounding water molecules; they bash into each other. All this bashing about breaks the weak bonds that kept the protein curled up. The egg proteins uncurl and bump into other proteins that have also uncurled. New chemical bonds form—but rather than binding the protein to itself, these bonds connect one protein to another.
After enough of this bashing and bonding, the solitary egg proteins are solitary no longer. They’ve formed a network of interconnected proteins. The water in which the proteins once floated is captured and held in the protein web. If you leave the eggs at a high temperature too long, too many bonds form and the egg white becomes rubbery.