Sure, Wiles obviously understands the main thrust of his proof. But one could argue that Wiles' result depends on lots of other results, which in turn depend on other results, and so on through decades and decades of work, ultimately going back to the foundations of mathematics. Neither Wiles nor anybody else can claim to rigorously understand all of it. You can imagine this as a tree, with Wiles' work as root, and his dependencies as ancestors, and so on. An error at a lower level of the tree could, in theory, invalidate the root node.

I do agree with Buzzard that it's hard to be sure. I've definitely read papers where a critical argument isn't well written or what is written seems wrong. However, if there are low-level errors, I suspect that with some work things could be patched up.

I think the intended implication is that Wiles has not read or understood all of the proofs he built on.

If someone said, "nobody dead or alive understands all the details of Windows" I doubt anyone here would find it controversial. It's too big to fit in one person's head.

Buzzard asked Wiles himself, who stated "that he would not want to sit an exam on the proof of Langlands–Tunnell".