For sure, string diagrams are meant as a tool for someone experienced with category theoretic proofs. It's possible that they could be used in introductory material too--a la that quantum book that's very diagrammatic--but this paper isn't it.
Category theory is kind of funny in that it is very simple at its base, but rapidly advances from there. Its greatest strength is how compositional it is, but that also means that most practitioners rapidly begin composing things at many levels of abstraction and it can get tough to follow.
String diagrams technically help with that. They have their own compositional rules which map to the ways category theoretic constructions compose, but they still look "simple" to the reader and can provide insight. But in all cases, you need to be comfortable working each component all the way through to fully grasp what's going on. And that can be pretty complex!
