There isn't any sure-fire method of explaining anything, and especially in math. But specifically in the case of the Monty Hall problem it has been proven by extensive experience that many individuals with otherwise above average intellectual capacities exhibit an exceptional tenacity in refusing to accept the (otherwise) widely agreed upon solution; don't waste your time on repeating this experience, unless you should find pleasure in doing so.
It should also be noted that for experts in probability theory there is a lot of nitpicking to be done about the exact hypotheses that need to be formulated in order to get a completely well defined probabilistic problem with the purported solution as correct answer (the Wikipedia page on the subject illustrates this). So if the resistance you get is of the type: "it is not so clear cut, it might depend", then very possibly they could actually be right, unless you took all the required precautions in explaining the problem.