You can define the start of an infinite series, just not the end. (Except as ∞ or -∞). You could also have an infinite set that extends both ways.

0 to ∞ contains an infinite amount of numbers. But so does 11 to ∞.

More filled rooms do not “appear”, the rooms just go on without end. These is no “last” guest who moves into some previously unoccupied room. It’s just… endless. Infinite.

It really only makes sense in abstract. Our minds aren’t built to deal with infinity.