Transgressive things go against the rules. Letting your dog romp around on the town tennis courts is transgressive, especially when the "No Dogs Allowed" sign is very obvious.
Being transgressive means you're violating a rule, law, standard, or expectation. Sometimes transgressive behavior is dangerous, like a transgressive driver who likes to speed through school zones. But often it's just unexpected, disturbing, and maybe even offensive, like transgressive art that seeks to shock its viewers. Either way, it goes beyond the usual boundaries; the Latin root, transgredi, means "go beyond."