From replies to the question "What is category theory useful for?" on Mathematics Stack Exchange, some metaphors are enlightening.

Asaf Karagila:

Generalization is the bread and butter of mathematics. The idea is that often there are many similarities between two seemingly unrelated objects, but their difference itself is very cluttering. It makes the analogy hide under the surface and not jump right at you.

When you abstract something by cleaning up the clutter you find yourself with highly concentrated theorems about tons of objects. Sometimes it would have been very hard to prove each case, let alone to come up with the idea to prove these cases. The generalizations can make things easier to see, easier to prove and easier to handle.

That been said, the de-generalization of going back from a very abstract theory to a more "down to earth" one (even if it is a very abstract theory on its own) can be a very hard process as well. Sometimes the abstraction process kills a lot of interesting information that you would like to know. Things like actual calculations or methods to calculate certain values. Restoring the details can be quite the hard work, but the abstraction helps us see what is true to begin with, and that's an incredibly important thing in mathematics, which is often similar to wading down a bog of peat, blindfolded with your shoelaces tied from one shoe to the other. Generalization and abstract theorems are lighthouses.

Ittay Weiss

Category theory serves several purposes. On the most superficial level it provides a common language to almost all of mathematics and in that respect its importance as a language can be likened to the importance of basic set theory as a language to speak about mathematics. In more detail, category theory identifies many similar aspects in very different areas of mathematics and thus provides a common unifying language. The fact that almost any structure either is a category, or the collection of all such structures with their obvious structure preserving mappings forms a category, means that we can't expect too many general theorems in category theory to be really interesting (since anything you can prove about a general category will have to be true of almost everything in maths). However, some general truths can be found to be quite useful and labour saving. ...

Continuing the analogy between set theory and category theory as common languages for maths, the situation is like saying "a set is just a bunch of points. No structure. Nothing. How interesting and relevant can it possibly be to study set theory?". As it turns out it is both interesting and very important to the rest of mathematics. ....

Less superficially, category theory encourages a shift in attention to what structure is. In category theory the particular inner workings of an object do not matter at all. All that matters is how that object is related (via the morphisms) to the other objects in the category. ...

A Ellett

At a certain level, mathematicians study structures. In a very small nutshell, category theory allows you to talk about structures recurring throughout other disciplines of mathematics; it's a kind of unifying theory. Much like Algebra studies various arithmetic structures and distills their properties without having to worry about whether you're talking about integers, matrices, families of functions, etc. which in the specifics are quite different, so category theory distills properties across various disciplines of mathematics. ...

(cf Cakes, Custard, and Category Theory (2016-02-14), 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), If You Need a Theorem (2018-11-08), ...) - ^z - 2018-12-24