Lets put this another way.

You and I are sitting and deciding if this is true. You decide that it doesn't matter, but I know it does. So we go to the game show and do the show a thousand times. At the end of the competition, I will have walked away with 50% winnings and you would of walked away with 33% winnings. That's reality.

The math is an abstraction of that outcome. It's confusing, but it is grounded in reality. The outcome, with enough iterations, is absolute. In fact, if we ran this simulation with 1000 doors, I'd still walk away with 50% prizes while you would walk away with 0.1% prizes.

It's simply visualized - take a piece of paper and do a binomial tree of the possible outcomes.

The reason it is so hard to grasp is because it runs counter intuitive. Your first choicedoesn't matter. Think of it as two seperate trials - in the first, you have to pick the winning 1 out of 3 or you have to pick the winning 1 out of 2. You always pick the 1 out of 2, if you want to win, so you essentially repick the only choice that is from the 1 out of 2.