Hilbert's Grand Hotel has a room for every natural number, all occupied. A new guest checks in anyway; so does an infinite busload; so do infinitely many infinite buses. The night manager's tricks are bijections, and they work until one bus arrives whose passengers cannot be listed — and a diagonal argument turns them away.
A finite hotel is full when the guests run out of rooms. An infinite hotel is never in that position: ask everyone to step one door down and the request never fails, because there is no last guest standing at a last door with nowhere to go. “Full” and “no room for you” come apart.
The other half of the trick is that a re-rooming is a rule, not a schedule. The manager does not walk the corridor knocking on doors; he announces “your new room is twice your old one” over the tannoy, and infinitely many people move at once. A single rule can do an infinite amount of work in one step — which is why the fleet of buses is no harder than the one bus.
Then a bus arrives whose passengers are infinite coin-flip sequences, and the manager’s luck runs out. Whatever list he posts in the lobby, you can walk down it building a passenger who disagrees with room 1 in flip 1, room 2 in flip 2, room 3 in flip 3 — and that passenger is nowhere on the list.