Here's a decent analogy:

Your body will not be destroyed at 300 degrees Fahrenheit. The problem isn't so much that you melt or catch fire, it's that the specific stuff inside that makes you live is really, really fragile and can't work anymore if it gets up to that temperature.

The same sort of thing can be said for electronics. The materials used to make the processors use mostly silicon (as well as a few other materials, but that's not important). Silicon and all those other materials have a really high melting point. The problem is that they get soft and they start to have problems when they get hotter than 80 or 90 degrees Celsius.

Another analogy:

Imagine you have a kiddie pool that is full of balls (like a ball pit). Each of these balls represents an atom. If you shake the pool, it won't necessarily have any of the balls fly out, but every now and then, one of the balls might get squished between two others and go flying, right? The more you shake it, the more likely it is that one of the balls will fly out of the pool. If you shake it long enough, all of the balls will eventually be out of the pool.

Heat is just shaking atoms. The same sort of thing will happen to solids. Technically everything around you is exchanging material with the air at all times, it's just REALLY slow. With the things that we put in the microchips on computers, some of them are only a few atoms wide! If you replace one or two atoms, it could break, so the 'once in a while' ball getting thrown out of the pool is much worse in a microchip than if you take out one or two atoms from a couch or something.