Why is the verb "weave" being used here? Is it because of "fabric"?
Yes, that's correct! "Weave" and "fabric" are words that go well together. The speaker does not mean them literally and has used them as a poetic metaphor to express the combination of different elements to create a new, innovative idea. Ex: The principles of multiculturalism are woven into the fabric of Canadian society. Ex: Colorful designs are woven into the fabric of the blanket.