The best answer is probably that certain enzymes and other proteins only work right at room temperature. That's why we mammals keep our body temperature in a narrow zone. That will probably just make you wonder why those proteins have to be at a certain temperature. Now things get complicated.

Enzymes are almost always made of protein. They have to be a certain shape to do their jobs, such as turning some kinds of molecules into other kinds. Enzymes make life possible by making the right chemical reactions happen.

Heat is movement of molecules; more heat means means molecules moving faster. If the molecules are moving too slowly, they don't meet up with each other often enough and all the necessary reactions don't happen. If they move too fast, they damage the enzymes so that they can't do their job. If the egg gets too cool, i twill slow down development. If the egg gets too hot the enzymes are destroyed and the egg can't develop at all.

This delicate temperature balance is one reason that almost all mammals keep their eggs inside. Why don't birds keep their eggs inside?

Most land animals have evolved to work best when they are warm. Cold-blooded animals (scientists call these 'ectotherms' which means 'outside warmth') like snakes and lizards have to get warm by sitting out in the sun before they do anything active like moving around or eating. Warm-blooded animals (we call these 'endotherms' - 'inside warmth') like you, me, and birds, can usually keep themselves warm, so we're ready to move around anytime and don't have to sit around in the sun beforehand. However, some warm-blooded animals do a better job of keeping themselves warm than other animals do. Really big animals tend to stay warm a lot better than really small animals. Think about babies -- when it's really cold outside, a mother will really dress up her baby in a big coat and hat and booties and a blanket and everything, because babies get cold really fast. But the mother will probably be wearing a much thinner coat and no blanket or hat, because she's a much larger animal and is much better at staying warm. The baby's father is probably even a little bit larger than the mother, so he might be even better at staying warm and might not need a coat at all.

So, back to your egg. Growing really quickly, like the embryo in an egg does, is a lot of work. The bird just can't do it unless it's warm. Now an adult chicken does a pretty good job of keeping itself warm -- it has plenty of fat and feathers and whatnot, and it can move around and move its muscles and make heat. But the little embryo in the egg is quite small and can't really keep itself warm. So the hen has to sit on the egg to keep it warm (or you have to put it in an incubator) in order for the embryo to stay warm enough to grow quickly and hatch.

So now you're probably asking "OK, so the egg has to be warm because birds have evolved to grow best when they're warm. Why haven't birds evolved to grow best when they're at room temperature?" This is a good question, because it seems like it would be a lot easier to stay at room temperature than it is to stay warm. Well, it turns out that chemical reactions happen faster at higher temperatures. So all the chemical reactions that go into eating and walking and flying, growing (and everything else birds do) goes faster when it's warmer. Since a bird that grows and eats and flies quickly will do better than a slow bird, being warmer is generally better than being cooler. There's a limit, though. If things get too hot, then the chemical reactions go so quickly that they start to get out of control. So if a bird stays too warm, all the reactions go too fast and all of its proteins and enzymes and everything else will start to break down. The body temperature that birds (and mammals and you and I) usually keep is about as warm as possible before things start to go haywire. So that's why it's best for eggs to be warm and not at room temperature.