A bipartite game graph has two sets of vertices: player positions and player positions. Edges only go between the two sets. This structure arises naturally in turn-based games. Player positions are where player moves, and vice versa. Bipartite structure simplifies analysis.
You know whose turn it is based on which set the current position belongs to. The bipartite structure means positions alternate between two sets. This often simplifies analysis because you can characterize winning positions by which partition they belong to.