I'd say category theory informs math in two main ways. First it is essentially the "language" in which many areas of advanced math are written. In algebraic geometry we could define sheaves without using terms from category theory but when we use category theory everything becomes much clearer. Similarly when talking about morphisms of sheaves or adjunctions, everything become clearer when we use the language of category theory. The same principle holds in algebraic topology, only more so. We couldn't really have done most of the past 60+ years of algebraic topology research without having category theory to do it with. Second, categories are interesting in their own right, and tie into other areas of math. For instance we can take the classifying space of a category to get a topological space, and this connection turns out to come up a lot in homotopy theory. For another example, the algebraic K-theory of a ring is actually about the category of finitely generated projective modules over that ring. So we can study algebraic K-theory of other categories, and do it using category-theoretic tools.
