Just after he tells her the first part of that, he adds, "as if I needed another motive to kill you." I think that he means that his revelation to Bella had the opposite effect of that - in other words, it made him NOT want to kill her even more strongly. It drew him to her, and when he realized that she had kept his secret, it made him see how special she was. He may have also been hoping that the more she learned about him, the more she would fear him; but it had the opposite effect on Bella, too. The more he confided in her, the more she loved him.
He fought with his family, but Carlisle, Esme, and Alice sided with him; they wanted him to stay and be with Bella, if that's what he wanted. The rest of the family thought it would be better for him to leave than to risk exposing them to the public. So he had to fight against his family and his better instincts to stay with Bella and to keep himself from hurting her, but he managed to do it because at that point he started to get to know her better, and he fell in love with Bella as a person and not just the call of her blood.