The Hotel With Infinite Rooms — All Full, Always Accommodating

A hotel has infinitely many rooms numbered 1, 2, 3, ... and every room is occupied. A bus arrives with infinitely many passengers — one for each positive integer. The hotel manager accommodates all of them without evicting anyone. Then a second bus arrives — with as many passengers as there are pairs of positive integers. The manager accommodates them too. Then someone points out that the number of rational numbers is the same as the number of natural numbers. How? And can the hotel accommodate one guest for every rational number?