Note that X ⊆ Y is equivalent to X ∩ Y' = ∅ where Y' is the complement of Y and ∅ is the empty set. And, as you already know, X - Y = X ∩ Y'. Using these facts, rewrite both sides.
You can also do this directly from the definitions. To prove " " show that "if x is in X then it is in Y".
Here, we want to prove so start "if x is in A- C" and try to conclude that x is in B. If x is in A- C then it is in A but not in C. But we know that so every member of A except those that are also in B. Since x is in A but NOT in C, it is in B.
To prove the other way "if then do the opposite "if x is in A- B" and show it must be in C.