It's because your original probability is 1 out of n doors. The rest of the doors is (n-1)/n. The host does you a favor by showing the doors in the other group not containing a prize. By switching you gain access to the (n-1)/n probability with the uncertainty of the losing doors removed. This is the favor the host did--he removed the uncertainty in the other group.
So, you have the original choice with 1/n probability and the switching choice (n-1)/n.