This would be a deal breaker for me. I would have dumped him already. Yes lack of sex does make him feel unloved, unattractive, unwanted. It can lead to depression and cheating. However, he should have talked to you about this ages ago and tried to fix it and then he should have left before trying to get with anyone else if you refused to fix it. What he has done is wrong and its not the answer. It doesnt solve your problems. It has just made them 100 times worse.
How do you know that this wasnt the third or fourth or fifth time when he got caught? What makes you think it was the first? He could have already cheated and there is no way for you to know for sure if he is telling the truth or not.
You are both to blame for the lack of intimacy. You should not have allowed it to get this bad BUT there is no excuse for cheating and I would kick him out if I were you
"Don't ask a question if you can't handle the answer".