FWIW, I think the best "applications" (if they can be called that) come from domains like algebraic topology and algebraic geometry, where some concepts come up so frequently that it's useful to formalize them in category theory. The homology proof of Brouwer's fixed point theorem [1] is one good example. Things like Eilenberg-Maclane spaces are another (really is most natural to think of them the thing that represents some functor).

I wouldn't say that you can't do this stuff without category theory, but it does make it easier / clearer. (Similarly, you can do a lot of geometry without coordinates, but coordinates definitely make some stuff easier / clearer.)

[1] https://www.wikiwand.com/en/Brouwer_fixed-point_theorem#/A_p... [2] https://www.wikiwand.com/en/Eilenberg%E2%80%93MacLane_space