Key observation: you lose when switching the door only if your first pick was winning.
Your first pick had 1/3 chance of winning. So switching has 1/3 chance of losing. Thus it has 1 - 1/3 == 2/3 chance of winning.
Your first pick had 1/3 chance of winning. So switching has 1/3 chance of losing. Thus it has 1 - 1/3 == 2/3 chance of winning.
The key is making it clear that switching doors will always switch you from Lose to Win and vice versa. Switching doors is equivalent to switching outcomes. That's the critical fact, IMHO.
Once that's accepted, the rest falls into place: there's a 2/3 chance that you picked a Losing door, so there's a 2/3 chance that you'll benefit from switching outcomes, and since you're guaranteed to switch outcomes if you switch doors, there's a 2/3 chance that you'll benefit from switching doors.
Keeping the same door means "I bet I got it right the first time", which is only 1/3 probable!