Nature only uses the longest threads to weave her patterns, so that each small piece of her fabric reveals the organization of the entire tapestry.
Dr Feynman is referring to the fact that if you can find a simple mechanism that explains a complex system, then you can expect that mechanism to work in other places, possibly throughout all of nature. You can explain the motion of the planets around the sun with Newton’s theory of gravity and then notice that that same theory works for humans being pulled toward the center of the Earth, and that our solar system orbits around the center of the Milky Way. Thus, an example of the “small piece,” to which he is referring, is gravity.
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Did you ever come across his “imagine the Gods are playing a game of chess” analogy? It’s real good: http://www.youtube.com/watch?v=o1dgrvlWML4.