Sets do not (at least never seen them) appear in applications. Sets/Categories are used forfoundations. Foundations is a branch of math that focuses on the foundation of all math, definitions and proofs that appear in other branches of math.

Here is an analogy.... Set theory is like Abstract Expressionism. People (the non-artists) think that art needs to be nicely drawn (like the Renaissance). In this case People (the non-math) think math needs to have applications. But art does not need to be nicely drawn and can look stupid. Similary math can have absolutely no application and look stupid*.

*)Like in, prove that if: x>y and y>z then x>z. It looks stupid but there is much more to it.