Variant of offline Turing machine
Let $S$ denote the languages accepted by such machines.  The question wants you to prove $S=L$.

When you have a complicated-looking task, it often helps to break it down into pieces.  So, a first helpful hint on how to break it into pieces: try to prove $S \subseteq L$.  Separately, try to prove $L \subseteq S$.  (If you still can't solve it, then you're able to ask a more specific question about one or the other of those.)