You have a Hotel with infinite rooms in it. An Infinite Bus shows up with infinite passengers. As each passenger comes into the lobby, you give them the next room key, so that everyone gets a room.
An hour later another Infinite Bus shows up. Oh dear. Now what will you do? You call the manager and he says no problems. What you're going to do is go to room 1, apologize and ask them to kindly relocate to room 2. Offer a complimentary breakfast by way of apology. Then go to each subsequent room and ask the inhabitants to move from their room number to 2n. Now that all of the current guests have moved to their new, even numbered rooms, you can put the second bus into the odd numbered rooms and everyone is happy.
As Standup Maths recently pointed out, the problem is that we think of infinity as "count until you get really tired of counting and then it's one more than that". Which is just not a workable definition and causes problems.