A lot of category theory is just about composition, how different structures compose and how compositions of those structures continue to be composable.
So in this case, GLA builds the theory of linear transformations from trivial/simple pieces and their various ways of being composed. It also discusses the mechanisms for proof and reduction (you noted how GLA is clear about whether a sub-computation is reused or not, it'll also be clear about how those two choices are equivalent).
So it's really not studying "just" linear transformation (which are, in finite cases, summarized by a matrix of numbers/field elements) but also the theory of their construction, manipulation, simplification. It gives you a rich language for talking about how two linear combinations might be related to one another, something that's more challenging to access from a matrix.
Well, that wasn't very concrete. How about this: can you give an example of a linear-algebraic fact that is more easily shown by reasoning about these diagrams than by just using the numbers-moving-along-wires interpretation to immediately turn it into regular equations?
For example, the only thing I can think of is maybe (AB)^T = B^T A^T.
So in this case, GLA builds the theory of linear transformations from trivial/simple pieces and their various ways of being composed. It also discusses the mechanisms for proof and reduction (you noted how GLA is clear about whether a sub-computation is reused or not, it'll also be clear about how those two choices are equivalent).
So it's really not studying "just" linear transformation (which are, in finite cases, summarized by a matrix of numbers/field elements) but also the theory of their construction, manipulation, simplification. It gives you a rich language for talking about how two linear combinations might be related to one another, something that's more challenging to access from a matrix.