A narrative is a description of a sequence of events, sometimes created or spoken as events are unfolding live.
It's often used in a manner synonymous with story, in which case the narrative will be pre-written or pre-planned and have an introduction and ending. So "pushing the narrative along" is something that causes a subsequent event in the story to happen. The writer of your example is saying the percussive score was needed to get the story moving faster, as it was too long and boring.
Note that lately it's become fashionable to use this word as a substitute for text/speech that supports/furthers an agenda, which implies the "story" is a real-life political battle and the "ending of the story" is a specific political aim.