Why should we expect the abstract notion of flipping arrows in a category to generate useful concepts from other useful ones? What exactly does flipping the direction of arrows mean and why is it universally useful? It certainly does not seem to be anything meaningful in any concrete category itself(or does it)?
Edit:I can see that duals are useful anyway since they generate free proofs but this would not answer my question.
The main question here is I suppose: Why would one expect dual concepts to be non vacuous before actually taking duals and checking?
There are cases where the opposite of a familiar category constructed by arrow-flipping is, or is equivalent to, another category of real interest (rather than an artificial construct): the opposite of the category $\mathbf{Set}$ is equivalent to the category of complete atomic boolean algebras (a surprise when you first hear it, perhaps).
But ok, suppose for the sake of argument that there are relatively few such examples: suppose that flipping arrows only rarely takes from an intrinsically interesting category to some other category we might want to investigate for 'concrete', non-category-theoretic, reasons. This perhaps is perhaps the thought underlying the question.
No matter! Arrow-flipping would still be a useful technique, giving us proofs by duality. For a baby example, if we show that in every category terminal objects are unique up to unique isomorphism, then -- remembering that categories come in pairs related by arrow-flipping -- it will follow by duality, i.e. by arrow-flipping, that every initial object is unique up to unique isomorphism. And so it goes.
The importance of arrow-flipping, i.e. of duality considerations like this, in producing buy-one-get-one-free proofs does not depend on the flipped version of particular interesting categories being themselves interesting. It only depends on categories having duals, interesting or otherwise.
And it can hardly be denied that the resulting duals of useful theorems are very often useful too! To take obvious examples, some familiar constructions are, in category theory terms, products, and other constructions are coproducts. It isn't that the one kind, products, is inhabited and its dual, coproducts, is empty! Likewise some constructions which are familiar pre-categorially turn out to be equalizers, other familiar constructions turn out to be coequalizers. Dualizing theorems about products and equalizers gives us theorems about coproducts and coequalizers -- and there are indeed familiar examples of such things! So yes, both the theorems about products and equalizers and their duals have interesting applications.