I have just "finished" this book and put my hands up in dismay at the thought of reviewing it.
Roger Penrose is now 85 and this book may be the final presentation of his worldview. Although he talks about the 'layman' as his audience, potential readers should recall "The Road to Reality". In Amazon reviews of that tome, retired maths professors and physics PhDs lined up to recount at which chapter they hit the limits of their knowledge and had to give up.
This volume is not so different.
In a nutshell:
1. Penrose dislikes String Theory because its extra dimensions admit too many functional degrees of freedom (basically the number of possible field configurations). It is not explained clearly why the super-explosion in the functional freedom space size is problematic, although he does make a related point that he believes that the six 'curled-up' dimensions are actually unstable and should collapse.
Perhaps it's obvious.
2. Quantum Theory is seen as a partial or incomplete theory - in particular, Penrose thinks that its linearity will be violated in an improved theory. He believes that the reason we don't observe 'Schrödinger's cat' spatial superpositions is due to the gravitational effects of superposition (he takes spatial delocalisation to have a real gravitational effect, aligning with his ontological realism for the quantum state). Specifically, the gravitational self-energy due to the superposition generates energy uncertainty, equivalent to time-uncertainty, hence superposed stationary states collapse into a position eigenstate very quickly. As he explains it, the maths behind this is pretty advanced, requiring general relativity.
3. Cosmologically, Penrose is not a fan of inflation, basing his criticisms on the 2nd Law and entropy. His criticisms have force suggesting that inflation retains support faute de mieux.
What does Penrose himself suggest as alternatives? He thinks twistor theory (a framework featuring emergent space-time) continues to have promise, and believes that a particular kind of bouncing, recurrent universe traversing through repeated big-bangs can explain the extraordinarily low entropy 13.8 billion years ago.
I think it's good for physics that he wrote this book, but absent a huge background in general relativity, complex analysis, twistor theory, quantum field theory and tensor analysis it's difficult to assess the merits of his arguments.