How can a Fields Medallist be 'not very good at logic'?

I think the sentiment expressed by the author is pretty clear: being a good mathematician is being able to go back and forth between being able to be really precise (logic), and being able to explore (and maybe even 'intuit') interesting relationships and connections, play with new ideas, speculate, hypothesize, etc.

But, as such, I do believe Sir Michael did (probably intentionally) overstate his case a bit: as he says himself, at some point you need to take these new ideas and suspected theorems and make them hard, and that requires being good at logic. So I agree with you there that this seems like a strange statement of his. Indeed, Sir Michael is of course perfectly capable in logical thinking!

Then again, he does refer to some of the high technical formalizations of logic, and how that seems almost too 'extreme', too 'restricted' ... how indeed it is hard to see how, say, a deeply formal logical axiomatization of, say, arithmetic, gives us any novel ideas about arithmetic (if you work with the Peano Axioms, I think you'll get the idea). So that kind of logic is something he says that he resists doing.

But in the end you're right: to say that he is not good at logic at all, seems overstated, though most likely intentionally so and for dramatic effect: I think it's his way of trying to get those of us who do tend to stay in the purely 'precise' domain to 'venture out' a little more!

One can be an excellent mathematician and be helpless in face of even the 1st order logic (aka predicate theory). Just take a look at https://en.wikipedia.org/wiki/Principia_Mathematica (admittedly, this is a bit extreme). I remember taking logic classes and having to face this staff: it felt completely foreign to me, even though it describes (to some extent) how mathematical proofs are structured.

If he's a mathematician then he's very good at the kind of logic that's taught in secondary school, which is the only part of logic that's used in actual mathematical proofs. The purpose of that kind of logic is to assure correctness of proofs.

Now suppose someone asks whether the ratio of lengths of proofs to lengths of theorems is bounded. That is also a question of logic. There are many questions of that kind, and that's probably what he was talking about.

PS: It's not bounded.