I'm not sure about "claims to". I have misplaced my copy of the Penrose book. Hofstadter's book is humorous, despite its weight. It doesn't purport to be a canonical explanation of the theorem.

Both books are ultimately focused on the nature of consciousness, although from quite different perspectives. In both cases, the theorem is presented as one element of an argument.

It's interesting that two sophisticated writers with wildly different opinions about consciousness have both leaned on Gödel to back their arguments. There's nothing in the theorem that shouts "This is about consciousness!"

Penrose's argument linking consciousness to Godel's theorems makes little sense to me.

https://en.m.wikipedia.org/wiki/Penrose–Lucas_argument

Thanks for answering my question. I am guessing that neither book would explaim the proof in detail.