The best explanation I've seen was in series 3, episode 3 of Jame May's Man Lab, where they repeatedly iterated the problem and showed that when repeated enough

The best explanation I've seen was in series 3, episode 3 of Jame May's Man Lab, where they repeatedly iterated the problem and showed that when repeated enough times, the frequency of winning/losing tends to a different value with a change of door than without (by having the presenters open booby-trapped beer cans and either get splashed with beer or not)