The following quote gives a good description of the value of abstract mathematics. The quote speaks specifically of “universal algebra,” but consistent with the spirit of the quote you could generalize it to other areas of mathematics, especially areas such as category theory.
Universal algebra is the study of features common to familiar algebraic systems … [It] places the algebraic notions in their proper setting; it often reveals connexions between seemingly different concepts and helps to systemize one’s thoughts. … [T]his approach does not usually solve the whole problem for us, but only tidies up a mass of rather trivial detail, allowing us to concentrate our powers on the hard core of the problem.
Emphasis added. Source: Universal Algebra by P. M. Cohn
Related: Applied category theory