“why would people even want oil back in that period? They didn’t have cars.”

You have a semantic problem here. You are thinking of "oil" in a modern sense of the term meaning petroleum, whileas your teacher meant vegetable oil, like olive oil. Olive oil was not quite a precious commodity, but something necessary, as it had many uses: cooking, elaborating preserves like escabeche, fuel for lamps, wound cleaning, etc.

This types of uses are documented in many books like the Llibre de Sent Soví (a medieval cookbook), the Book of Hunting Animals by Abraham of Toledo, the anonymous Book of Horses, and many documents between private individuals.