Imagine you have one person for each (positive) integer, given a unique integer ID at birth, and a hotel with countably infinite rooms, each with a unique room number.

The hotel could have someone in everyone room if every person with an even number as their ID was staying there. That is, for every room number, twice the room number is a unique even number, so there's a 1-to-1 correspondence between the number of rooms and the even integers.

You could then have someone with an odd number ID show up looking for a room, and would have to rearrange from a stay-in-half-your-ID lineup.

It's a (arguably defining) property of infinite sets that they contain a strict subset (at least one guy not in the subset) with the same "size" as the whole set.

So the evens and the integers are an example of this, with both having the same "size", even though the evens are contained in the integers.