"I don't even teach very well.", what does "even" mean here?

As @mythociate says, it is a comparative adverb. Its original sense must have been to indicate that something is 'equal to', 'the same as' or 'like' something else (as opposed to being 'more than' or 'less than' something). It is now used rather in the sense of adding some information to an existing statement, often to say that something which might be expected to be true is not ("Even though I speak English, I cannot understand what he is saying.", "I can't play the piano; I don't even know how to read music.") or to emphasise or extend a comparison ("The profit this year was even more than that made last year"; "He gave me more, even, than I had hoped for.").

The speaker is comparing 'how well he teaches' to 'how well he does something else,' it sounds like. Like maybe someone asked him if he can speak to a classroom full of students, and he said 'Speak to students? "I can't even teach very well."' "Even" in that usage is a comparative adverb. I guess I would need a little more "context" to know for-sure. But I think--though the sentence doesn't NEED 'even'--'even' connects it to the passage the sentence is in.

It may in here to emphasize the word "teach" and one more question. Are you living in Shanghai?