@Paige0,
He is probably thinking of the part of the Moderate Cancellation policy where 50% of unused nights can be refunded.
When he left, did he cancel or modify his reservation?
If not, then Airbnb has no way of knowing he left early, and will not issue any refund.
You also really don't know if he left the day before.
Inform the guest that the policy he is thinking of only applies if the guest either cancels or modifies the reservation. Also tell him Airbnb is the one that handles all money issues, so he will have to take it up with Airbnb support.
It would probably be a good idea to contact Airbnb support yourself to let them know what is going on with this, to make sure they know what happened before the guest starts talking to them about it.