How do I communicate to a Teacher Assistant that I would like to fix our relationship? I am a final year undergrad student in mathematics. This is an important detail as usually there are not many mathematics students, and even less in their final (because of the dropout rate). Because of this, the number of students is pretty small, around 12-18, which allows the teachers to notice every one of us, learn our names, habits and etc.

Now, one of the Teacher Assistant seems to dislike me. One of the reasons for this is at the beginning of the semester I spoke too much according to him. I took this criticism with no problem, and made sure to talk as little as possible during the classes, that is I spoke only when a classmate would ask a question based on the problem we were working on.
Yet, he still seems to be irritated by me. I once answered my friends question in a quiet way, to which my teacher responded: "you're such a pain". He likes to ask questions first to me, which no one is able to answer. In a situation where my answer was far from being right he responded: "you would better not answer than give such an idiotic answer". This isn't very detailed, and shows only my point of view, but I wish to fix this situation or at least improve it. I am unsure if this Stack Exchange is suitable for this question, in case it's not, I will delete my question.

Now, I would like to talk to him as I want to fix this situation and  to do so I would like to first send an email asking to meet up. I want to know what exactly should I state in the email? I definitely don't want this to backfire me in any way.

So my question: how should I phrase my email to ask for a meeting to discuss this situation while staying polite and very tactiful?

I will accept any advices concerning the handling of this situation.
I disagree with the posted answers and so wanted to weigh in. 

It seems you have become the problem student in this TA's mind. I have witnessed this many times, and the pattern seems exactly as you have laid out. The student is overly informal and talkative/particapatory in class. It disrupts the flow of class, making the lecturer lose concentration and taking the conversation down tangents which are typically not relevant to the course direction. Then, rather than have a reasonable conversation after class with the student, the TA or lecturer tries to shame the student into submission by snarky comments during class. Then, because two wrongs don't make a right, both parties are aggrieved. At that point, the TA has labeled you as a "problem" and any even minor infractions are dealt with harshly. 

If I wanted to repair the situation as a student, I would stop by office hours and begin by honestly apologizing to the TA for making their job more difficult. Listen to their response and repeat your apology if necessary. Then explain the efforts you are making to rectify the situation and ask if there is anything else you can or should be doing. Listen. Finally, offer an overall explanation for your behavior that might help the TA to see you in a different light. Probably your initial behavior was motivated by misplaced eagerness rather than malice, and if the TA can understand this they can perhaps reframe your current actions in their mind and respond differently. 

Yes, this puts the onus on you to fix the situation. However, I think that is somewhat fair, given your behavior initiated the situation in the first place. Hopefully the TA recognizes your olive branch and extends one of her/his own. Good luck.