In particular, the Gracchi violated an unspoken and unwritten compact that governed the behavior of the Roman aristocracy. In particular, they attempted to secure power for themselves using avenues not considered "in bounds". There was no institutional mechanism in place to process violations of that unwritten compact (social opprobrium had worked for hundreds of years), so the senators (Scipio Nasica in particular) immediately transitioned to personal violence.
Once the taboo against aristocrat-on-aristocrat violence vanished, Rome descended into waves of high aristocrats raising private armies to secure their personal power. It was, more or less, a game of last man standing that Augustus "won".