Injecting a sense of narrative into a game can be as simple as combining uncertainty with inevitability. People like to find stories in events, and inevitability helps them see stories in the games they play; if the game is drawing closer to a defined end point, players will naturally begin to see a narrative in the way the game approaches its conclusion. Uncertainty keeps the excitement going while that story plays out.
However, those aren’t the only tools designers have to create narrative in games. For example, stories (in some traditions, at least) follow a three-act structure: an initial period of exploration and preparation, a second act with lots of conflict and danger, and then a headlong rush toward the final reckoning. A game’s mechanics can be used to imitate that structure, such as by organizing cards so that they have noticeable quantum leaps in the danger they pose to players, or by situating resources on a map so that there are natural stages in the players’ conflicts over them.
You can even try something weirder. At one point this semester a group I was in considered a design wherein whole new mechanics would be introduced to mark act changes. That particular concept didn’t make it into the final game, but I think it’s a neat concept, and it’s something I’m sure I’ll return to as a way to ratchet up tension and create a sense of progress as a game goes on.
About mid-semester I was told that a game seemed “flat”–every turn was like the last until the game ended. In games as in soda, flatness is bad. 😉 Create drama and narrative by making it clear that the game is advancing toward a conclusion, keep the result uncertain until that conclusion is reached, and do whatever else the design permits to bring the feeling of a story to your games.