To understand Perelman's proof of the Poincaré Conjecture, you need a solid background in Riemannian geometry. Many books can be used for an introduction to this field. There are two books I like on this subject: Riemannian Geometry, by Gallot, Hulin and Lafontaine and Riemannian Geometry by Petersen.
After, you can try to learn about Ricci flow, a good starting point is Chow and Knopff's "The Ricci Flow: an Introduction". It covers the basics of Ricci flow including Hamilton's theorem that on a compact 3-manifold with $Ric>0$, the (normalized) flow will converge to constant curvature.
Then, if you want to go into Perelman's work, there is the book "Ricci Flow and the Poincaré Conjecture" by Morgan and Tian. However you also have to understand Thurston's Geommetrization Conjecture, so you need a solid background in 3-manifold topology,
I don't know the references for this part, maybe Thurston's lecture notes?
Another interesting road is to study the proof of the differentiable sphere theorem by Brendle and Schoen, a good reference is Brendle's "Ricci Flow and the Sphere Theorem".
I hope that was helpful.
First, let me make some preliminary remarks. We sometimes like to think that "being proved" is a black-and-white property, but in fact there are shades of gray. At one extreme are things like the infinitude of the primes, whose proof every mathematician understands. But then there are results that are widely accepted but no proof has appeared. Vladimir Voevodsky has pointed out that "a technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail," and that this practice can lead to false statements being erroneously accepted as proved; recognizing this point led Voevodsky to spend much of the later part of his career on computer verification of formal proofs. Although the mathematical literature is generally very reliable, it is far from perfect; this point has been addressed in more detail in another MO question about the extent of wrong papers in research mathematics. So in
vast majority of cases, "being proved" isn't about being 100% confident
that there is no error; it's about whether the proof has been sufficiently
scrutinized that the chances of a serious mistake are negligible.
Returning to Yau,
if you look carefully at what he is saying, you will see that, technically, he does not say that he thinks Perelman's proof is wrong, or that it has a serious gap, or even that there are parts of the proof that nobody understands. He says only that he is not certain that the proof is totally nailed down, and that he has not met anyone who understands the most difficult part of the proof. He also points out that if a powerful new idea is properly digested by the mathematical community then it usually leads to the solution of new problems, and that if this has not happened with the most difficult part of Perelman's proof then it probably means that this part of the proof deserves more study.
In principle, calling for the mathematical community to devote more time to studying an important and difficult proof in order to "nail it down" and acquire a "complete understanding" and a "full command of this work" is unobjectionable. In the past, I have heard colleagues say that the original work of various Fields Medalists—Hironaka and Freedman come to mind—was very difficult to understand and that there was a need for the community to study and assimilate those groundbreaking ideas more thoroughly. In both the cases of Hironaka and Freedman, the community has indeed put in effort to study their work, and rich dividends have resulted, so this type of activity is definitely worth encouraging. Note that this doesn't mean that the original proofs were wrong or had serious gaps; it just means that the proofs moved closer to the infinitely-many-primes ideal of universal understanding, and the chance of an unnoticed significant gap or error was driven down even closer to zero.
Unfortunately, Yau chose to phrase his remarks in a "politically fraught" manner that he knew would "get him in trouble."
He says things in a way that (probably intentionally)
gives many readers the impression
that he is casting doubt on the correctness and completeness of Perelman's proof (even though, as I said, technically he doesn't explicitly say that the proof is wrong or incomplete). The book appeared in 2019 but the most recent conversation he cites was from 2006.
He makes no mention of recent research in the area which
does in fact apply Perelman's ideas to solve new problems.
It should therefore not be surprising that the consensus of the mathematical community is that Yau's remarks do not pose any serious challenge to the conclusion that Perelman's proofs—especially of the Poincaré Conjecture, which involves fewer technicalities than the Geometrization Conjecture—are correct. There were at least three separate efforts which came to this conclusion. Kleiner and Lott's detailed notes say, regarding Perelman's original papers [51] and [52]:
Regarding the proofs, the papers [51, 52] contain some incorrect statements and incomplete arguments, which we have attempted to point
out to the reader. (Some of the mistakes
in [51] were corrected in [52].) We did not find any serious problems, meaning problems
that cannot be corrected using the methods introduced by Perelman.
Similarly, Morgan and Tian wrote:
In this book we present a complete and detailed proof of the
Poincaré Conjecture. … The arguments we give here are a detailed version of those that appear in Perelman’s three preprints.
There is also the account of Huai-Dong Cao and Xi-Ping Zhu, which Yau himself refereed.
On top of those three detailed accounts of Perelman's proof, there have been more recent developments. Terry Tao mentions the recent survey by Richard Bamler. Moishe Kohan mentions Kleiner and Lott's Geometrization of Three-Dimensional Orbifolds via Ricci Flow and Bamler and Kleiner's proof of the Generalized Smale Conjecture. So contrary to the impression you might form from what Yau said, the community is indeed continuing to milk Perelman's ideas and apply them to solving new problems. If there are specific technical points which Yau thinks are obscure, I am sure that other researchers would be happy to address them if Yau were to spell them out explicitly. Until then, there is no credible reason to doubt the fundamental correctness of Perelman's arguments.
Best Answer
Most likely, this is just misremembering (or misattribution).
In his math writing Thurston did not make any predictions regarding what approach to the Geometrization Conjecture (GC) will be successful. I did not hear him making such predictions in his math lectures (but, of course, I heard only few), and I would be surprised if he did. What Thurston might have said to somebody privately, I do not know.
However, Thurston was well aware of the power of the Ricci Flow (RF) and in the early 1980s Thurston did write few pages with a sketch of the proof of the Orbifold Theorem which is the geometrization theorem for orientable irreducible orbifolds with nonempty orbifold locus:
W. P. Thurston, Three-manifolds with symmetry, preprint, Princeton University, 1982, 5 pages.
One of the steps of the proof was an application of Hamilton's 1982 result on manifolds of positive curvature (proven via RF, of course). A proof of the Orbifold Theorem was given later on in a work by Boileau, Leeb and Porti (broadly speaking, their proof follows Thurston's outline), shortly before appearance of Perelman's preprints. A bit different proof was later given in a work of Cooper, Hodgson and Kerckhoff. Both proofs use Hamilton's theorem at some point, as Thurston suggested.
Kapovich, Michael, Hyperbolic manifolds and discrete groups, Progress in Mathematics (Boston, Mass.). 183. Boston, MA: Birkhäuser. xxv, 467 p. (2001). ZBL0958.57001.
I summarized several known approaches to the GC. Two approaches were differential-geometric (RF and the Mike Anderson's approach), one was via min-volume (a combination of differential and Alexandrov geometry) and one was coarse geometric (to the "hyperbolic part" of the conjecture). At the time, it was quite possible that the GC will be proven in a piece-meal fashion with one proof of the hyperbolization conjecture, another for the Poincare Conjecture and yet another for Spherical Space Forms Conjecture.