Penrose's basic construction is to connect a countable sequence of open Friedmann–Lemaître–Robertson–Walker metric (FLRW) spacetimes, each representing a big bang followed by an infinite future expansion. Penrose noticed that the past conformal boundary of one copy of FLRW spacetime can be "attached" to the future conformal boundary of another, after an appropriate conformal rescaling.
Furthermore, there does seem to be some experimental evidence. Specifically:
In 2010, Penrose and Vahe Gurzadyan published a preprint of a paper claiming that observations of the cosmic microwave background made by the Wilkinson Microwave Anisotropy Probe and the BOOMERanG experiment showed concentric anomalies which were consistent with the CCC hypothesis