During expansion, dark energy -- the unknown force causing the universe to expand at an accelerating rate -- pushes and pushes until all matter fragments into patches so far apart that nothing can bridge the gaps. Everything from black holes to atoms disintegrates. This point, just a fraction of a second before the end of time, is the turnaround.
At the turnaround, each fragmented patch collapses and contracts individually instead of pulling back together in a reversal of the Big Bang. The patches become an infinite number of independent universes that contract and then bounce outward again, reinflating in a manner similar to the Big Bang. One patch becomes our universe.