I've said before that math isn't actually done in the abstract formal proofs you find in papers. A mathematician will noodle around with some models (often inspired by some application!) until some problem finally yields, and then he rewrites the result in a formal style that makes it sound like he was following a straight logical path to the answer all along. There's probably no help for that, but filling out papers with some more examples would help understanding. Certainly help me, anyway.
If his posts are any guide, Mumford seems to like examples. The most recent (as of today) is "An Easy Case of Feynman's Path Integrals." "Easy" is perhaps a bit relative, but I like the step-by-step way he evolved his model. Good writer.