Student or Learner
It reminded me of ''work one's fingers to the bone'' but one of other users has already mentioned it. I think ''pull up your socks'' can also be used in the same sense. Can't it?
The idea, I believe, in ''work your socks off'' is when you walk harder or run, or play which involves one or both of them your socks get low/lower (hope I can convey what I mean) and you can be called to have worked your socks (nearly) off.