At some point, someone must have looked at a self bow and said "mmmm..." But how many writings do we have from the rank and file bow builders from any culture? I think it evolved, much like any other invention, in power, sophistication, etc.
The crossbow is ancient, and as Orcbow indicated, emerged from China, the same guys who brought us gunpowder. The crossbow had to have been developed for the same reasons it was loved by Medieval generals. It is much simpler to train a crossbowman to shoot well, much faster than it takes to train a bowman. It would have been more expensive, though, at least for self bow cultures, though composite cultures would have spend a great deal more coin on arming their bowmen.
Dane