"To move the story" means to advance or progress the narrative or plot of a story. It refers to the actions, events, or developments that push the story forward and contribute to its overall progression or resolution. Full definition
A profoundly moving story of simply very decent people refusing to change the scope of their humanity to fit the times they were living in. (religion.blogs.cnn.com)
It eliminates the need for text to be the sole method of moving the story forward. (joystikrevolution.com)
At no point are there any slow points and every scene fits the purpose of moving the story along. (hidefninja.com)