> Like if I notice that a digrammatic language can be formalized as a hypergraph category, what do I learn about the network/diagramming language?

The main thing you instantly learn is that you can stick together diagrams in a whole bunch of ways and get diagrams that make sense. This 'whole bunch of ways' is the definition of 'hypergraph category'.

The second thing you instantly learn is a whole bunch of equations saying 'if you stick together diagrams like this, and then stick on some more like that, then you get the same thing as if you did... something else'.

The third thing you instantly learn is that the semantics of your diagram language should be a hypergraph functor: in other words, a functor that's compatible with all these ways of sticking together diagrams. That's indeed how it works in all the examples I've studied so far.

Furthermore, Brendan's theorems on decorated corelation categories give recipes for actually defining choices of semantics that will be hypergraph functors. So, you don't need to do this 'from scratch'.