Here's a way to think of it that some people fine helpful. Assume N boxes, the host knows where the prize is, always reveals N-2 goats after the content picks leaving just the contestant's box and one other box unopened, and always gives the contestant a chance to switch after revealing the N-2 goats.
A person who does not switch wins if and only if their initial pick was correct. The probability of that is 1/N where N is the number of boxes.
A person who switches wins if and only if their initial pick was not correct. The probability of that is 1-1/N.
Switching is (1-1/N)/(1/N) = N-1 times as likely to win.
A person who does not switch wins if and only if their initial pick was correct. The probability of that is 1/N where N is the number of boxes.
A person who switches wins if and only if their initial pick was not correct. The probability of that is 1-1/N.
Switching is (1-1/N)/(1/N) = N-1 times as likely to win.