[Continued]
Besides discussing with your boyfriend about the causes and misconceptions behind his problems, the best way to help him is to accompany him and discuss on the subjects where he feels insecure.

While discussing with him, show and train him how to properly evaluate his performance: If he does something well, don't tell him he's "good", or "smart", but explain to him why you think that what he achieved is good (and get his opinion), and be specific, no generalities. If he fails or under-performs, get him to explain and discuss what did not work and why, and force him to give up on explanations based on his lack of worth by concentrating on the concrete causes of failure. (BTW, if one day you have kids, never compliment them on their worth, but on the worth of what they did. Same goes for the bad things).

Get him to speak out his thought process, when he discusses his abilities, and correct him as you go. Ideas like "I'm good", "I'm not good" or "I'm not good enough" should be banned from his thought process. He should think "this was good because of [this specific reason]", "this was not good because of [that specific reason]". If you think like that you know what to change when you fail, and what to keep when you succeed.

Also, never again accept that he tells of him that he's stupid or you smart; he should always give context: I did this poorly because of this and that, or you did better than me at that because of etc...

Beware: you will push him out of his comfort zone (otherwise he would not change), so it's likely that he'll turn his discomfort against you. Don't take it personally, and don't let it escalate. When it happens, give him some slack, and try a different approach. Otherwise he'll develop the habit of attacking you as a new defense mechanism.