Why does Bilbo refuse the treasure at the end of the adventure?

Bard does offer Bilbo a significant amount of treasure at the end of the adventure, saying to him, "I would reward you most richly of all." However, while Bilbo does not actually refuse all the treasure that is offered to him, he does not accept the share he might originally have had a claim to. He says that this is because he doesn't know how he would have managed to get the treasure back to the Shire "without war and murder all along the way." The journey is still a long one for Bilbo, and the treasure would have made him conspicuous. Moreover, he says he doesn't know what he would have done will all that treasure if he ever did get it home.
What this says to the reader about Bilbo is that he is not a greedy person. On the contrary, he is genuinely good at heart and content with his lot. He lives in a nice hobbit hole which duly satisfies all his needs, and he cannot imagine wanting huge amounts of treasure simply for the sake of having it. In the end, Bilbo does accept a chest filled with silver and a chest filled with gold as his recompense for taking part in the adventure—and, of course, he also has with him the ring he took from Gollum.