The story that Asher tells Jonas that changes Jonas's life is about a man who was unhappy in the Community, and resolved his problem by "(going) out and jump(ing) into the river, swim(ming) across, and join(ing) the next community he came to". The story is significant because of two reasons. The first reason is because it is the first time Jonas seriously considers the idea that a person might not find his niche in the Community, and the second reason is because it presents to Jonas the possibility of moving beyond the realm of the only community he knows.
Asher brings up his story during the Ceremonies, when the Twelves are anxiously awaiting their life assignments. He says he once "heard about a guy who was absolutely certain he was going to be assigned Engineer...and instead they gave him Sanitation Laborer". Terribly unhappy, the man made his escape the next day, and "nobody ever saw him again". Jonas is skeptical about Asher's story; he tells Asher that "somebody made that story up". Jonas has never heard of "anyone...who joined another community", and so does not quite believe what Asher is saying, but Asher's story makes Jonas start thinking, and plants the seed of thought leading to his own desperate escape attempt at the end of the book (Chapter 6).