I understand this intuitively.
Let's say there's $x \in A$ and $x \in B$. Then $A - C$ means we're taking away every element in set $C$ from set $A$ and similarly the same for $B - C$.
I understand that even if $C$ and $A$ had the same elements, even then $A-C \subseteq B-C$ holds true (the empty set).
But I can't seem to write a formal proof out for it.