Bilbo gives them the arkenstone to aid bard in his bargaining with Thorin. Bilbo knows how much the Arkenstone means to thorin and believes if the bard uses it to bargain with thorin it can help prevent a war. Bilbo doing this also shows to the bard and elvenking that he is an honest thief and it helps them relax and trust Bilbo because they were suspicious of him at first
