I feel there is an important insight here, but somewhat hidden. Let me try to capture it in one domain: mathematics.

Any mathematician will tell you that the way to prove good theorems is NOT to start with the axioms and move down the tree of implications until you hit upon something that looks interesting.

Instead, the way is to start with examples, or with an intuition, and try to formalize it into a conjecture. And then try to build further intuition as to why it might be true; possibly by trying more examples or building some heuristic arguments. Once you get a sense of why it might be true, then use that intuition to look for techniques that people have used to capture that intuition. As an example, if you feel that the objects you're studying have some notion of closeness, then you can introduce a topology on your objects and then use the techniques of topology to make further progress. And only at the end, when you're almost sure about how it's going to go down, do you build rigorous proofs starting from some simple statements.

So I guess Chesterton is trying to emphasize that building an intuitive, heuristic understanding of why something might be true is much more important than trying to build a deductive argument using logic. The latter always follows the former. I would be very interested in examples outside of math.