To mean business is to be serious about what one is promising or proposing to do, and the speaker is indicating that they let people know that they are serious by biting back.
Consider the staetement "I bite back, not enough to do damage but definitely enough to let him know that I am serious about what I am saying.". Based on this statement, is "I bite back, not hard enough to do damage but definitely hard enough to let him know that I mean business." a true statement? Yes/no/unknown