Another reason to care about Category Theory, from a comment by Aloizio Macedo in an online math discussion:

What point am I trying to make? The point is: the important property that categories are really abstracting is structure itself. Category theory endows a mathematician with a capability to soar through the fields of mathematics with a broad vision. More than "inheritance of properties", there is identification of a common underlying structure. The naturality of such identification comes with time and experience.

And all the above is an understatement. The distillation process which category theory provides is a powerful tool in several aspects. For example, it is common to have a situation where we want things to exist and they are imposed by some condition. And the imposition commands the definitions, and proofs are matter of routine checking. ...

*(cf Greatest Inventions (2011-06-09), Cakes, Custard, and Category Theory (2016-02-14), Category Theory Concepts (2016-04-25), Bird's-Eye View (2016-07-20), Category Theory for Programmers (2017-05-12), Ultimate Abstraction (2017-08-24), Put the Vast Storehouse in Order (2017-10-04), Yoneda Perspective (2018-10-03), Category Theory is like a Lighthouse (2018-12-24), Macro vs Micro (2019-02-03), ...)* - * ^z* - 2019-03-22