> Could you go into more detail on the examples you mention, where doing so led to new insights?

Good question. It's a bit hard to do so (especially without going into mind-numbing technical detail) -- in math you never quite know where insights really come from. "Fortune prepares the prepared mind."

But generally speaking, I would say that good exposition gets you thinking about: Why does the technique work? What is the key insight? What are its limitations? And if you think about such questions, you naturally get a better sense of for which other questions your techniques are also likely to work.