The phrase "to finish the story" means to complete or conclude the narrative or account of something that has been told or written. It symbolizes bringing closure or resolution to a particular storyline or situation. Full definition
You might even finish the story mode in this game before you get a bunch of them. (gamechup.com)
I also think that the game would've benefited from a better and more well thought out multiplayer mode, in order to keep crowds coming back after finishing the story. (metacritic.com)
Too many first time writers just don't finish the story in a satisfying way. (lulu.com)