One possibility is that they get inspired by the fanfiction. Maybe there's a story where the characters have to face a new and powerful enemy that they haven't encountered in the original series. This could give Team Avatar new ideas on how to improve their own skills and strategies. Also, if the fanfiction focuses on their relationships, like a romantic subplot between Zuko and Mai that's different from what they know, they might start to think about their own relationships in a new light.
Well, they might have a whole range of reactions. Aang could be really excited to see different interpretations of his story. Katara might be a bit critical if the fanfiction doesn't do justice to her character. Sokka would probably find some parts really funny and make jokes about it. For example, if there's a fanfiction where Aang tries to learn a new bending technique in a really strange way, Sokka would surely laugh and make fun of the whole situation.