Wording on Wall of Fire 5e

Hello so I have been running a Curse of Strahd with a group of friends. Been going great, but I have one player who usually likes playing rules lawyer alot. In a recent session he used Wall of Fire against a horde of enemies. Which is smart and I was going to have the enemies give the wall a wide berth and go out another cave entrance.

The player however said they take damage from the 10 foot radius which greatly confused me because they aren't entering the literal wall of fire, they are avoiding it. After some time I just gave in because it was a large battle anyway. I looked up the spell later to clarify for future sessions since I remember thinking to myself that sounds broken.

Sure as shit, it specifies in the spell that if a creature ends it's turn in the 10 foot radius, then it takes damage. Yet the player keeps insisting that they have to take damage. Please I need everyone's input, because it is greatly frustrating.