Reference introductory books or articles to Game Semantics
I know three introductions that may satisfy your criteria. All of them will use some Category Theory, but the first one (by Jürjens)  is actually quite light on that side -- he actually defines what categories and morphisms are in the context of game semantics, on page 111 (11th page of the article). You may want to read this one first, and then the others.


Games in the Semantics of Programming Languages -- an elementary introduction by Jan Jürjens.
Game Semantics by Abramsky and McCusker.
Semantics of Interaction (arXiv:1312.0121v1 [cs.LO]) by Abramsky.