Here's what he hears:
* The dwarves are planning a journey to the Lonely Mountain to reclaim their treasure stolen by Smaug the dragon. Thorin explains their lineage and their claim to the treasure. He describes Smaug's hoard, including the Arkenstone, which he wants to keep for himself as the rightful king.
* Thorin mentions needing a "burglar" to help them get past the dragon's defenses. They have heard of Bilbo's daring adventure in the Troll's cave, and think he might be the right person for the job.
This conversation sets in motion the entire adventure of "The Hobbit."