"You tell me a story."
Translation:Mihi fabulam narras.
"me" is not right.
"ad me" is not right, it means "to me" with a move. Going to me...
Your confusion comes from the English language, where you can say "tell me/her a story", it could look like a direct object first.
But you tell the story to someone.
So, it's the dative. You cannot use "me".