I don't get how exactly that would work though, since there would be an infinite cycle of universes. I don't see how we could be here if there was an infinite amount of universes before us, when does our turn come?
This isn't my field, but even if the universe is known to cycle repeatedly, this is simply our cycle in a chain of cycles yet to come. Keep in mind that each cycle is purported to take several hundred billion years, based on the rate of current blue-shifting and the minimum temperature before gravity itself fails.
Multiverse theory fills in the hole that we currently have with the "start" of expansion. When there's a gap in matter (the 'light' matter that we can see, and the 'dark' matter that is very difficult to perceive), the space is deemed 'dark energy', which is this force that holds universes together. Unless there's been some big discovery recently that I'm not aware of, there is currently no means in current science that allows us to perceive dark energy, much less know a use for it. POINT IS, expansion keeps happening, and the universe gets incrementally colder. Well, multiverse theory states that collision of these expanding universes is what causes expansion (big bang).