What is Bilbos explanation for why he gave the Arkenstone to Bard and the Elvenking? How does Thorin react to this?

Bilbo gives the Arkenstone of Thrain, 'the Heart of the Mountain', to Bard in order that he may use it as a bargaining chip with Thorin who treasures it. Bilbo tells Thorin it was done to broker peace, now that the treasure has been recovered. Thorin, although not impressed, actually agrees to this.