I will prove to you that God exists

[Angrboda]

In this video Rees details that while his speculations about the constants preceded the theorems of inflation, it wasn't the work on the constants but rather the ideas and theorems concerning inflation that led to the supposition that there might be multiple universes, and that these universes might have different constants and laws.  He makes the point that inflation and string theory both play a role in the suggestion of a multiverse.  It was not his work on fine-tuning that led to it.  He says as much in the book that Drew cites, Just Six Numbers, which I will quote as well.

"[B]ack in the 1970s we speculated about this [fine tuning] and I wrote a paper with Bernard Carr ... I don't think at that stage we had any partcular view as to whether these counterfactual universes might be a consequence of a real physical theory but the development of the idea of inflation and the realization that our universe might go far beyond the observational horizon did lead one to speculate that may actually be places ... the inflationary idea was developed in 1980s and that led to the possibilities that there could be domains and at the same time the realization came about that perhaps different big bangs might as it were cool down differently, it was the idea that the laws of low energy physics in the world we inhabit may be the outcome of so-called phase transitions as a universe cools down and these may have happened differently in different universes and this leads to the idea that what we call universal laws might be as it were environmental accidents in our cosmic patch and not the deepest level of universal laws.  So I think the idea that there might be other big bangs became discussed widely after the development of inflation, and there have since then been other ideas involving other space times."





"This ‘multiverse’ concept, though speculative, is a natural extension of current cosmological theories, which gain credence because they account for things that we do observe. The physical laws and geometry could be different in other universes, and this offers a new perspective on the seemingly special values that the six numbers take in ours."

Martin Rees.  Just Six Numbers.

In the book you refer to he fields the hypothetical that universe had to be as it is. He rebuts the hypothesis because it begs the question why if a universe has to be in a certain configuration, why is it the narrow configuration that causes planets, star, galaxies, molecules and ultimately life to exist. It doesn't solve the problem only pushes it back. That's why he continues to believe in multiverse theory.

In Just Six Numbers, Martin Rees does not use the phrase begging the question a single time. I'll need an actual quotation before I comment on this.

Given that I've just shown you misrepresenting Martin Rees, it's not a stretch to imagine that you are doing so again.

(Other words Martin Rees doesn't use as you've used them here: 'solve', 'problem', 'push', and 'back'.)

If the underlying laws determine all the key numbers uniquely, so that no other universe is mathematically consistent with those laws, then we would have to accept that the ‘tuning’ was a brute fact, or providence. - MR