This is because:
* The Arkenstone's value: The Arkenstone is incredibly valuable, and its possession would make the Elven King and Bard powerful figures. They realize they can't keep it without facing the wrath of the Dwarves and potentially starting a war.
* Bilbo's safety: They understand that Bilbo is a hobbit and not equipped to handle the potentially dangerous situation that could arise if he stays.
* The Dwarves' claims: The Dwarves have a rightful claim to the Arkenstone, and the Elven King and Bard see it as the best course of action to avoid conflict.
This moment is crucial to the plot of "The Hobbit" as it highlights the conflict over the treasure and sets the stage for a peaceful resolution.