Let's say the prize is behind door number 3. Now let's examine the "switch" strategy using all 3 scenarios.

Scenario 1: I choose door number 1. Monty opens door number 2. I switch to door number 3. I win..
Scenario 2: I choose door number 2. Monty opens door number 1. I switch to door number 3. I win.
Scenario 3: I choose door number 3. Monty opens door number 1 or door number 2. I switch to the other door. I lose.

Don't look now, but I've just won two thirds of the time! It's trivial to see that if I had not switched, I would have only won in scenario 1 – one third of the time