Penrose puts forth an old idea, that the end of our universe is the start of a new one, in a beautiful new way. That is, eventually the universe will lose track of the scale of space and time. So the whole giant universe turns into a Big Bang of about zero size!
Astronomers have recently found out that the expansion of the universe is accelerating, so it doesn't look like it will ever recollapse in the ordinary way. But in Penrose's theory, this apparently depressing reality is essential for a cyclic universe, because it means that the universe at the end of time is a spacelike surface, so its geometry can match the geometry of the Big Bang singularity in the next eon.
He offers a partial answer for an old puzzle: why did the Big Bang have such incredibly low entropy? The second law of thermodynamics tells us that entropy never decreases. The low entropy of the Big Bang defines the arrow of time, since entropy has been increasing since then, and life wouldn't be possible without a low-entropy state to start from. But where did the very low-entropy Big Bang come from?
Penrose's answer (or part of it) is that black holes destroy the information that goes into them (whether black holes destroy information is a big controversy in physics). That means that when the black hole eventually evaporates by Hawking radiation, the entropy that was in the matter that fell in has been permanently destroyed. I don't know if this can destroy enough entropy to explain the next very low entropy Big Bang.
Penrose doesn't believe the inflation theory, which is that space expanded incredibly rapidly right after the Big Bang. He says his conformal cyclic cosmology theory explains the things that inflation was invented to explain: it explains correlations in temperature in the cosmic microwave background between areas that are separated by large angles, and the scale invariance in the temperature fluctuations. The CCC theory also requires Weyl curvature to be zero at the Big Bang. This apparently explains why we don't see magnetic monopoles, another thing that inflation is invoked to explain, although Penrose doesn't discuss this in his book.
The CCC theory seems much more appealing than the inflation theory. It's more parsimonious, not requiring extra fields or an incredibly rapid expansion of spacetime. The universe would have expanded at the normal rate, only over a very very long time before the Big Bang.
The big hole in Penrose's theory is that our universe can only lose track of the scale of space and time if rest mass disappears. Rest mass gives a scale to spacetime. So it's necessary that all particles should eventually decay into massless particles like photons, or lose their rest-mass some other way. He hasn't come up with any good explanation for how this would happen. His best attempt at a theoretical framework for the decay of rest mass is:
"A standard procedure for addressing the idea of an 'elementary particle' is to look for what are termed the 'irreducible representations of the Poincare' group'. Any elementary particle is supposed to be described according to such an irreducible representation. The Poincare' group is the mathematical structure describing the symmetries of the Minkowski space M, and this procedure is a natural one in the context of special relativity and quantum mechanics. The Poincare' group possesses two quantities referred to as Casimir operators, these being rest-mass and intrinsic spin, and accordingly the rest-mass and spin are deemed to be 'good quantum numbers', which remain constant so long as the particle is a stable one and does not interact with anything. However, this role of M appears to be less fundamental when there is a positive cosmological constant L (Greek letter Lambda in the book) present in physical laws (as L=0 for M), and it would seem that, when we are concerned with matters related to cosmology, it should be the symmetry group of de Sitter space-time D, rather than of M, that should ultimately be our concern. However, it turns out that rest-mass is not exactly a Casimir operator of the de Sitter group (there being a small additional term involving L), so that its ultimate status is more questionable in this case, and a very slow decay of rest mass seems to me to be not out of the question."
I don't know how convincing this is. Does rest mass need to be a Casimir operator of the spacetime, to be a good quantum number, so that it's conserved for a particle as long as it exists? Apparently nobody's worked out what becomes of quantum mechanics and the Standard Model of particle physics in de Sitter spacetime. Until they do, and rest mass really does turn out to fade away in the expanding universe, Penrose's theory will limp badly.
Perhaps the above quote will tell you whether you'll find Cycles of Time to be readable, or whether it'll make your eyes glaze over. I found it readable, but I'm somewhere between the "intelligent layman" and a real expert. I loved Penrose's earlier book The Road to Reality, and worked all the exercises in it that looked challenging, except for one.
In Cycles of Time, unlike in Road to Reality, Penrose relegates almost all equations and mathematics to a couple of appendices, where he explains the transition from the scale free geometry at the end of the previous eon, to the Big Bang. The dynamics of the earlier universe propagate through the Big Bang. There's a loose end: unwanted freedom in the spacetime metric right after the Big Bang, so it isn't fully determined by the universe before the Big Bang. Penrose proposes various ways of eliminating this freedom.
Penrose tends to throw around technical language without much explanation. Road to Reality might be a good reference; or Google it. He mentions "gravitational degrees of freedom" early in the book. What concretely ARE gravitational degrees of freedom, I wondered? I asked online, then noticed he defines them later in the book!
Even though he uses technical language, he's very good at making advanced physics accessible. I found Road to Reality to be intellectually nurturing. It stimulated me to learn multivariable complex analysis, which he uses in his work.
Penrose is a maverick who disagrees with much of the contemporary physics consensus. He dislikes many contemporary physics theories that are science-fictiony or kludgish, like string theory with its extra dimensions, and inflation. But as Mark Twain said, "Whenever you find yourself on the side of the majority, it is time to reform (or pause and reflect)." Penrose has done it again in this book: come up with a wonderful and beautiful idea for Penroseland, his mirror of reality. Perhaps his mirror focuses better than the consensus mirror.