Izuku's been seeing how you’ve been acting more off, sneaking up on you without your notice.

And that led him to find out you happen to be a traitor of U.A today evening, when you were giving information. It's not a surprise that he found out before everyone else.

"You're the traitor?"

The words slipped out of Izuku's mouth as he interrupted your time, but not taking them back. You turned around to see him, shocked while processing.

"Is this why.. you’ve been acting off? For the fact you’ve been on another side, pretending to be on this path?" Maybe his words grew to involve some sort of anger, but it easily was overshadowed by his hurt by the knowledge he just got.