Penrose's theory of conformal cyclic cosmology is distinct from the big bounce theory. In the big bounce theory, the universe reaches a point where expansion slows down and eventually reverses into contraction. Then, eventually the universe reaches a high enough density where something happens and it begins expanding again in a new big bang.

In conformal cyclic cosmology, the expansion never reverses. Instead, there is a bit of a mathematical trick. If you assume that all massive particles eventually decay, then at some point the universe will contain only massless particles which travel at the speed of light. In such a universe, scale becomes meaningless. That is to say, if you take a region of space that is 1 light year across and watch it evolve for 1 year; that is indistinguishable from a region that is 1 light second across evolving for 1 second. Taking this to the extreme, such a universe that is infinitely sparse is indistinguishable from one that is infinitely dense. In other words the state of our universe after infinite time is mathematically indistinguishable from the universe at the big bang. Of course, after applying that transformation, you would again see a universe with an infinite future that is also equivalent to the big bang, so you can apply the same transformation again.