The best starting point, for a reader who knows a little about both game theory and category theory, is the preprint Compositional game theory.
Additional background and motivation is provided by the blog post A first look at open games and the preprint Compositionality and string diagrams for game theory.
By far the most complete exposition is my PhD thesis Towards compositional game theory. It is fully self-contained for readers who know category theory but not game theory.
If you don’t have background in category theory, my current recommendation is Seven sketches in compositionality by Brendan Fong and David Spivak.