How do I communicate to a Teacher Assistant that I would like to fix our relationship?
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.