Long story short, the Big Bang is a singularly unique singularity which is mathematically no different than the massively expanded universe in the far, far future. Because they are the same, one infinitely expanded universe becomes the infinitely small start of the next.
The mathematics he uses to demonstrate this comparison is called conformal geometry, a math that remains consistent despite working with the cosmically infinite be it infinitely huge or small. Conformal geometry has some advantages, apparently, in that it "squashes" the infinities at the beginning and end of the universe into quantised concepts. It also has some advantages because it allows the Big Bang to occur without the need for Inflation in the very early moments of the universe.
In fact, one of the main theoretical arguments in favour of CCC is that it overcomes some issues with Inflation which the cosmic background radiation map presents, namely some ripples which should not exist with Inflation. CCC explains those ripples as the gravity outcomes of collisions between super massive blackholes or the final 'pops' as blackholes eventually evaporate in the previous universe. Gravity, the mysterious non-force force, traverses from one universe to the next and makes itself apparent in the ripples of the microwave background. He makes predictions about what those ripples would looks like and has a few people supporting him with claims that they see the suspected ripples.