How does my TOEFL score affect my odds of being granted a TA in US Universities? I'm an international student BTW.
Note that TA is normally only granted to a (usually) doctoral student. You need to be accepted into a program before you can be considered to be a TA and that will be determinative.
The evaluation will be different at every university as we have no national system here. Also, TAs do different sorts of things in different fields and in different universities/departments. Some actually teach undergraduate courses and some simply grade papers. In between are those common cases where a TA will work a few times a week with a small group from a large lecture section. For most of these situations fairly good English skills are needed, but each department will do its own evaluation. But it is probably understanding and speaking ability, more than writing, that is the most important thing as a TA.
Treat TOEFL as important but there may be other considerations that are treated as more important in a given situation.
My impression is that most departments / universities simply set a minimum score on the TOEFL (or other standardized exam) to be a teaching assistant. Usually this minimum will be stated in the application information. If you have at least that score, you'll be considered for an assistantship if you are admitted, along with all the other eligible candidates. If you have a lower score, then you won't.
It could be that some departments give preference to candidates with higher scores, but I am not sure how common that is.
At my university in the U.S., "The Graduate School" sets a minimum TOEFL speaking score, which is slightly lower than my math dept's minimum TOEFL speaking score for TA's and/or RA's. Further, there is an intense orientation program for non-native English speakers before the beginning of classes, and unsatisfactory effort or performance there results in a kind of "probation", requiring further English-speaking classes.
Most applicants with a less-than-minimum TOEFL speaking score will not be considered at all, because they'd be unable to do the job of TA. In some cases, faculty "lobby" for students with at-the-margin speaking scores, but must (in effect) promise that they'll guarantee that the student works seriously to improve their English, and possibly promise to support them as an RA indefinitely, so that English skills matter somewhat less.