He says he did, evidence says he didn't.
Option A: He didn't and is telling a lie or he has a false memory.
Option B: He did and his arrest didn't result in a charge or conviction, or it was swept under the rug thanks to his father's money, power, and influence.
Knowing how rich families operate and knowing Biden was planning on a political career early on in his life, Option B seems the most likely to me.