It's annoying that you need so much math to understand the utility of Category Theory. I learned a bunch of Category Theory before I ever saw it used in a useful way.
Grothendieck wrote modern Algebraic Geometry in the language of Category Theory. This is the first time I saw Category Theory really used in a useful way. Grothendieck's proofs of the Weil conjectures I would say is a good example of using Category Theory to solve a famous problem. Category Theory is used to define and work with etale cohomology and etale cohomology plays a fundamental role in Grothendieck's proofs of the Weil conjectures.
Grothendieck wrote modern Algebraic Geometry in the language of Category Theory. This is the first time I saw Category Theory really used in a useful way. Grothendieck's proofs of the Weil conjectures I would say is a good example of using Category Theory to solve a famous problem. Category Theory is used to define and work with etale cohomology and etale cohomology plays a fundamental role in Grothendieck's proofs of the Weil conjectures.
https://en.wikipedia.org/wiki/Weil_conjectures