Assuming he can be believed, and we might note that there's no benefit to lying about choosing to steal, you change the game so the other guy's payoffs are zero

Assuming he can be believed, and we might note that there's no benefit to lying about choosing to steal, you change the game so the other guy's payoffs are zero for stealing and 1/2 the money (as modified by the chance he's telling the truth about that). So it changes the incentive from wanting to defect, to either getting nothing or choosing to trust him and maybe get something. Note also that he was going to steal at first, but changed his ball at the last second to cooperate. And the other guy lied about stealing, having chosen to split the whole time, never changing his ball.