All it requires is that the universe be infinite in size. If you go in any direction for a certain distance (and they calculate the distance) you will HAVE to encounter a (subset of the) universe that is identical to this one.
They start quite simple. Imagine a universe in which there are only four atoms: A,B,C,D and only four locations for those atoms to exist: 1,2,3,4.
There are only a certain number of ways you could lay out this grid before you had to start repeating a pattern. This would happen in any direction you chose to go. You can derive a formula that says "with X atoms and Y spaces you can go only Z distance in any direction before you are forced to repeat the pattern of 4 atoms in 4 spaces".
(I'm not sure how they get around the obvious flaw in the argument: Who says the rest of that infinite universe has to have any atoms in it at all? Then you could repeat hard vacuum indefinitely without ever duplicating a pattern of atoms. I think there's some condition about large-scale homogeniety.)
Anyway, when you scale it up to universe size, the numbers (while very large) are not infinite. I think the number was something like 10^120 metres.
This means that, given the premise of a universe of infinite size, you could travel (let's say) 10^120 metres in any direction of your choosing, and land your spaceship next to an identical copy of yourself.
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