Why should I learn modern category theory if my interest mainly is structured sets? A long time ago I studied mathematics at the University of Stockholm. I had a romantic view of modern algebra and manage to make the first two algebra courses by self studies in order to immediately study homological algebra, Galois theory and such topics. That is not the best way to study. Later as a graduate student I did rather well - until the gaps in my basic knowledge and abilities began to affect too much. Then I stopped focusing on mathematics about 35 years ago.
I did self studies in category theory because we were supposed to do that and because it was a good idea. Category theory worked fine with the mathematics evolved at 1950 or so. The universal definitions and duality simplified a lot of mathematics as tensor products and injective/projective modules etc and the functors opened new possibilities.
The last 40 years or so the interest in and the development of category theory has exploded and seems nowaday be very abstract but also very consistent.
My question is, what modern category theory could be    interesting for a person mainly interested in the mathematics concerning structured sets?
The bounty will soon expire and there is 50+ in reputation to earn - aren't there anything to express on this topic?
 A: Perhaps the most compelling answer to the question:

My question is, what modern category theory could be interesting for a person mainly interested in the mathematics concerning structured sets?

is just that a great deal of modern algebraic geometry, as a result of Grothendieck and others, makes heavy use of category theory. Here, I'm basically replacing "structured sets" with "rings, fields and modules." However, the study of groups and monoids has also been immensely impacted by category theory. In particular, category theory has played a really large role in representation theory. 
If "structured sets" also includes topological spaces then there's even more category theory that ends up being relevant. And of course, topological spaces, and the study of their invariants, are important in things mentioned above like algebraic geometry and representation theory.
One problem with your question is that it's immensely broad. You're basically asking "How is category theory used in algebra?" And well, the answer at this point is almost "In what cases is it not used?" 
My suggestion, if you want to have some sense of what category theory is all about, is pick up Saunders Mac Lane's book and force yourself to learn the foundations (e.g. categories, functors, natural transformations) from a purely formal point of view, and then read about the examples. Then pick some topic you like (e.g. ring theory, or representation theory) and ask a more specific question about category theory in, say, representation theory. Again, for any of this to make any sense at all, you'll have to have a pretty good grip on whatever topic it is you're interested in. Category theory tends to produce "large scale" structural theorems, and so if you're not familiar enough with a topic to be interested in how all of its pieces fit together, it (in my opinion) will be very hard to motivate category theory. 
However, after writing all of this, and then reading through the comments above a bit more, I see that someone has really already provided you with an answer, which is this MSE question, so that's probably a pretty good place to start.
