I think Jim Dixon's "brute force" method is eminently clear and should be sufficient to convince the most mathematically naive that switching increases the odds of winning from 1/3 to 2/3.
Increasing the number of doors in the thought experiment usually makes the result seem more intuitive, but I think it may be more effective (psychologically) to consider increasing the number only to ten; observe that the "not switching" strategy wins the prize one time in ten., while the "switching" strategy wins the prize nine times in ten.
Or in the three door problem, consider that switching turns a correct initial guess into a loss but turns an incorrect initial guess into a win. Since the initial guess is twice as likely to be incorrect as correct, switching is twice as likely to win the prize.
If you're intrigued by thought experiments that seem to defy intuition, you might lose some sleep over Newcomb's paradox.