take this action lightly, or chooses to give the stone away in some misguided effort to anger Thorin; rather he bestows the gem upon the Bard and the Elven-king in hopes that the Bard will be able to use it as a bargaining chip, enabling them to end the stand-off between the dwarves, men, and elves.

Bilbo points out that "winter is coming on fast," and all the involved members of the conflict will be dealing with "snow and what not and supplies will be difficult--even for elves" (243).  To say that Bard and the Elven-king are surprised by Bilbo's actions would be an understatement, and they cannot help but question his motives. 

Bilbo explains himself simply. Besides his worries about the weather, he informs them:

"Personally, I am tired of the whole affair.  I wish I was back in the West in my own home, where folks are more reasonable.  But I have an interest in this matter--one fourteenth share, to be precise" (243).

The hobbit firmly believes that without some outside involvement in the negotiations, that Thorin is "quite ready to sit on a heap of gold and starve" (243).  Bilbo gives the Arkenstone to Bard, out of a sincere desire to help.  Later, Gandalf, who secretly listened to the entire exchange, commends Bilbo's actions, saying: "There is always more about you than anyone expects!" (245)

Your question touches on a key point in the story and is another example of the heroic nature of Bilbo. We see when the company arrives at the mountain that Thorin in particular becomes obsessed with gold-greed but also with finding the Arkenstone - an incredibly valuable jewel that is very important to his race and family. When news comes that Smaug is dead Thorin determines to protect the dragon hoard and not give any of it away, even if it means war and death. Bilbo is not taken in by this greed and can see Thorin's failing in this regard. He decides to bring this imminent catastrophe to a peaceful conclusion by taking the Arkenstone and giving it to Bard as a bargaining chip - he knows that Thorin will do anything to get the Arkenstone and thus he realises he can force Thorin to part with some of his treasure to gain the Arkenstone.

Of course, it is important to realise how this represents a further development in the character of Bilbo. Note how he is not now saving his own life as with his encounter with Gollum. He is also not now saving the lives of his company in the incident of the spiders and the escape from the halls of the Elvenking. He is now preventing death and war between three races through his heroic actions. This incident then represents the development of Bilbo from comfort-loving Hobbit to hero who saves others through his exploits.