Chemistry! Other than that... honestly, I'm not sure what the exact processes are. A quick google search: The reason why the eggs turn solid when cooked is because they are made up of lots of proteins. When the proteins of a raw, liquid egg come into contact with heat, it causes them to change form. The reason they change is due to the heat that makes the individual units of protein link together differently into what are ultimately stronger chains.
That's interesting. A kind of self-organizing matrix of proteins then. Chains linked to chains resulting in a sunny side up miracle.
From what I read, this is an example of a curing process: http://en.wikipedia.org/wiki/Curing_(chemistry) The proteins in raw egg white are long chains curled up into balls. Heating make the balls uncoil, and then the unravelled long chains form chemical bonds from one to another, thus linking them all into a bound network. Some glues and resins do a similar thing though often initiated by something than than heat. The key thing, as Kittimaru commented originally, is that this is a chemical change rather than a simple change of physical state.
*nods* It's funny... for as much as we (as a species) know about the world around us... we know precious little... and even less about the "why" of so many things.
From article: "Pouring egg whites into a beaker of acetone will also turn egg whites translucent and solid." That's interesting, though I wouldn't exactly serve it for breakfast.
And as you may know, you can "cook" fish by marinating it in lemon juice (e.g. ceviche) or vinegar (soused herrings etc). Very tasty, too. So proteins can be denatured by a variety of means.