It's mostly just a way of communicating the bizarre nature of infinite series and other problems related to infinity. Just fun thought experiments.

It's not always discussed as such, but Hilbert's Hotel is a mathematically well defined topic and can be proved rigorously. An infinite set of rooms can be the set x1, x2,...xinf, and people can be y1, y2...yinf. You can pair every entry in these two sets. x1&y1, x2&y2,...xinf&yinf. You can't number a room without having a person in the room, and you can't find a person who doesn't have a room.

It has to do with countably infinite sets.

The analysis on Wikipedia does a better job of explaining the concept: https://en.m.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel#Analysis

The whole point is that it's something we can prove mathematically that is highly unintuitive.

If you have n rooms and n guests there are no empty rooms. Let n go to infinity and there should still be no empty rooms.

The trick of the Hilbert hotel is that if you add a guest to a hotel with countably infinite guests the number of guests does not change.

The hotel should really just fill its even numbers rooms first, then if another person turned up after you already have an infinity of guests, you just house this start of a new Infinity in room 1 etc