In "the game" men and women compete for immense prizes, but the prize for losing is always horrible
Some of the typical prizes include large amounts of money, houses, sex and on special occasions, slaves. Men often compete for a chance to impregnate a woman since the women who compete are often chosen for being attractive. "the game" is made up of multiple events, but these vary from episode to episode and so does the theme.
game mode will improve your experience with the story, but it isn't strictly necessary for most branches.