Why does body sweat smell funny?

By stale sweat I'm assuming you mean sweat that has dried? First off, there are lots of electrolytes that you lose through your sweat. Secondly, bacteria like to grow in moist areas and sweat is very moist as you well know. When bacteria get in the sweat, they use it as a food because you have all these nutrients you lose and when they use the sweat as food, they release a gas and this is what gives it that smell. The smell will stay even after it is dry because it has not been washed away.