To "break a record" means to get a higher score or a greater accomplishment than anyone else in history. For example:

Phelps broke seven world records at the 2009 Olympics.

This means that Michael Phelps swam faster than anyone in history for 7 different events.

Bands often like to come up with unique names that don't really mean anything. So the meaning of "Recordbreaker" (a person who breaks records) doesn't have to relate to the band or its music.