How to persuade my advisor to keep working with me even if my grades are poor?
scaaahu has given the most pragmatic answer in a comment:


  Study harder or leave. 


If it is very important to your advisor that your grades are better, those seem to me pretty much the options.

That being said, I think most people don't really care so much about grades that they would kick out a perfectly fine grad student for grade's sake alone, even if he is well above the formal requirements. Honestly, I would assume that this is not, or at least not only, about grades.

How is your research going along? Are you progressing at least as well as the majority of the other students in the lab? Is your advisor still enthusiastic about your topic? It is not entirely unlikely that your supervisor is not super-happy with some other part of your work, and just using your suboptimal grades as a simpler, more objectively validatable reason to put pressure on you.