The impact of the quote in this context could be two things.
First, it could refer to the ramifications or significance of the quote on the rest of the story. Marc Antony's speech to the mob after Caesar's death in Julius Caesar, for example, was a pivotal moment in the play. It completely altered the political dynamic in Rome, leading to the destruction of Brutus and the other conspirators. So a quote from Antony's speech could be said to have an enormous impact on the story.
But it also could refer to the emotional, dramatic, or other effect that the quote has on you, the reader. Good writers (i.e. the type whose works we read in English classes) carefully choose their words for the impact they will have on the reader. This could be a quote from a character, like Macbeth's bleak speech just before he went off to fight Macduff in the final, climactic battle of the play, or it could be a particularly pithy or powerful line by the author, like Dickens' immortal opening to A Tale of Two Cities: "It was the best of times, it was the worst of times." Of course, many quotes that have a significant impact on the reader are also important to the story.