Neither that article nor Wikipedia's article on string diagrams does a good job of explaining what they are. Afact, they are dataflow diagrams (aka circuit diagrams) dressed up with category theory mumbo-jumbo.
That is of course, a joke, but Category Theory does feel like that sometimes. There does not appear to be any grounding, it's all circular. Anything can be lifted-up, or projected forgetfully down, to almost anything else. A Category has sets of nodes and edges, to describe the category of ... SET. Category diagrams are directed graphs, and there is a category of GRAPH, which can be labelled to make a diagram, which can be used to analyze ... categories.
There is always sand under your feet and concrete in the sky.
A string diagram represents a computation using a monoid (associative binary operator with a common left/right unit element). You can interpret the computation as a dataflow through the inputs and outputs of the operators. By further characterizing members of the set, you can add constraints on the types (parts) of the data flow along edges.
So they are just a monoid algebra, dressed up with category theory mumbo-jumbo: monoidal category.
The origins go back to Einstein Summation Convention for writing products in vector/matrix/tensor math. Albert got fed-up with writing all the correct free-bound indexes and ranges on fancy sigma summations, so he invented the lazy summation convention.
To make a dot product of two vectors they must have the same dimension. To multiply two matrices, the i-dimension (no. columns) of the first argument must equal the j-dimension (no. rows) of the second, and these will be consumed, then the result will retain the other two dimensions.
Not only are matrix multiplications non-commutative, but the output dimensions will be different depending on the order. For example, A32 x B23 = C22, but B23 x A32 = D33, note how the outer dims must be equal and are consumed, and the result has the inner dimensions.
Penrose was a specialist in GR, so he knew the summation convention in his bones. He was also a very visual mathematician, who had appreciated Feynman's diagrams as a planar graph representation of terms from a mathematical series expansion of the particle interaction. Penrose developed a diagram notation for tensor operations:
All this algebra can be categorified, expressed in the language of CT. The various behaviors of commutativity of the product give rise to a spectrum of categories (symmetric, braided, etc.), and these can be represented as properties of the diagrams. For example, braiding means the order is important, and the crossing of two edges in the planar graph has an algebraic meaning.
The best layman's introduction to how this ties together various parts of math, quantum physics and even computer science is: Baez, Stay "Physics, Topology, Logic and Computation: A Rosetta Stone" (2009)
The theory and practice of applying string diagrams to quantum mechanics has been taken forward by Bob Coecke. Here are a couple papers that go through QM math, categories and string diagrams:
