It drains a portion of the opponent's Barrier Gauge if they Barrier block it. The amount of Barrier drained varies from character to character, and can be increased by holding A+B.
They are also very useful in combos and are often used mid combo. Some character's Crush Triggers force a hard knockdown, some ground bounce, others float the opponent very high. Whatever the case, they are all useful to help extend combos and deal more damage.
Crush Triggers ignore character combo rate and have Same Move Proration, so it is best to only use one per combo.