It's wrong to reveal one's secrets just to spite them.
Anthony had recently told Billy a deep dark secret about himself, and a few days later they are now in a large fight in front of their other friends.
Billy wants to win the argument with Anthony.
Billy uses something Anthony said prior in the argument to prove he is wrong.
Billy uses the secret Anthony told him against him, to try to get points with everyone else.
Anthony continues arguing for a while, but eventually concedes the argument.
Anthony is devastated, and runs off, never to talk to Billy or the rest ever again.