If you're not familiar with this concept, there is a typical flow stories should follow that can be summed up with something called Freytag's pyramid (or Freytag's triangle). To sum it up briefly, the story should start (exposition), build action (rising action) up to a high point (climax) and then fall back down (falling action) and close up everything nicely (denouement).
When thinking about this for video games, I also started to ponder how a denouement would work for board games. Typically, someone flips a card, rolls a die, or moves a token and declares victory and that's that. I guess the falling action takes place during the cleanup (or setup of the next game) while people discuss what they could've done better or how they should have won. In other games the falling action is playing for second, third, fourth, etc. place. If you want your board game to tell a story, you should definitely keep this triangle in mind.