I'm normally not too sympathetic to the "abuse excuse," but please let me explain why I think Jesse's case is different.
He didn't just "come to" and decide that he "may" have been abused as a child. There were pictures of him with a black eye and a broken arm, both administered by his father who, if I remember correctly, was his primary caregiver. Having been continually abused by someone who should have been your protector does do something to you. He was continually verbally and emotionally abused as well. We tend to believe our caregivers as children. If we are told - or treated as though - we are worthless and don't deserve good things, we tend to buy into it. It sounds like it wasn't until he got into rehab/therapy that he realized how self-sabotaging his adult behavior was.
I was similarly abused as a child. Then I married the kindest, sweetest person in the world, and then began to try to make the self-fulfilling prophecy of "you don't deserve someone this good" come true. I was not very nice to him. I was pushing him away before he had a chance to reject me first. Finally, after 13 years, he looked me right in the eye and said, "No matter how you act, you can't make me stop loving you, so why don't you stop trying." What a life-changing moment that was. If there was one thing I knew about him, I knew that he didn't say things he didn't mean. That was when I started consciously working on changing how I behaved to him and to others. We have now been married for over 45 years, and the last 32 have been a lot better than the first 13.
We've all seen celebrities (and others) use the "abuse excuse" in the hopes that people will cut them some slack for their misdeeds. But I've never seen anyone who also seemed to recognize their own responsibility in those misdeeds as much as Jesse did in his interview. He had nothing but wonderful things to say about Sandra, and he knows he blew the best thing that ever happened to him.
I guess the proof will be in the pudding. If Jesse does, indeed, refrain from similar "self-sabotaging" behavior in the future, the sincerity of what he said will be proven. I'm hoping that he really has learned. Call me a cockeyed optimist.