In the end of The Hobbit, he decided to only get about 3 small pouches of gold.
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It was laid out in the letter they left on the mantel. Bilbo was to receive up to one 14th share after expenses, the cost of travel would be defrayed. And any funeral expenses would be paid as well.
Because during the unexpected party, Thorin promised Bilbo 1/14th payment of the treasure in return for Bilbo's services as a 'thief'. Though Bilbo did not do much in the way of being a thief, he did save the dwarves from Thranduil, as well as help them gain entrance into the mountain. They also wanted him on the journey because it would have been unlucky to take on their journey with 13 members. At the end of their adventures, Bilbo had obviously earned his 1/14th share, so Thorin fulfilled his promise and sent Bilbo home with a chest of treasure (though it is hinted later on in LotR that Bilbo ended up receiving more).
He was thankful towards Bilbo as he had just received the greatest treasure in that can be found in the Lonely Mountain and was now able to try make a deal with Thorin Oakensheild for a share of the treasure to rebuilt lake town.
He had borrowed several handkerchiefs from Elrond. He had complained of having left home without any of them. Bilbo left him 'such small gifts as he would accept' when they stopped on the return trip.
Well, as he justifies it to himself, he was entitled to a one fourteenth share of the treasure and which fourteenth was not specified. Though he was pretty sure that the Arkenstone would not be a part that he would get.