In Hip Hop culture the break is that part of the record where everything drops out and it's just beat, like a drum solo. It was that stripped down funk that made the original B-boys (break boys) go off. Kool Herc was the first DJ to play two of the same record, switching back and forth between records to extend the break, giving the b-boys more time to dance.
But ask Kool Herc and he'll tell you that "break-boys" didn't come from breaks on the record. "Break" was a street term that was used in different ways. If you were breaking, it could mean that you were acting crazy, or causing trouble. It could also mean that you were going through a breaking point, which describes the situation of many living the the ghetto. They applied this exaggeration to the dance. Yes, b-boys dance to the break of the record, but they are called so because of the street term.