Long story short, aspiration for leadership / dominance.
By having bullied someone (and not having been told where to get off), the bully effectively establishes dominance over the victim, which indulges their ego and makes them feel strong and lifeworthy.
Anybody with power or strong character can become a bully towards less strong/powerful ones, unless educated or motivated not to.
That said, bullies do not necessarily have anything common in their childhood histories that caused them to become bullies, however, having been bullied in the past or having seen bullying increases the likelihood of one becoming a bully. Vivid examples of this are dedovshchina and ragging — where people justify their bully behaviour by "I was bullied by older mates before, now it is my turn to take it out on the newcomers", or simply seeing that as a de-facto tradition.