Why does Thorin give Bilbo a coat of Mithril?

Because during the unexpected party, Thorin promised Bilbo 1/14th payment of the treasure in return for Bilbo's services as a 'thief'. Though Bilbo did not do much in the way of being a thief, he did save the dwarves from Thranduil, as well as help them gain entrance into the mountain. They also wanted him on the journey because it would have been unlucky to take on their journey with 13 members. At the end of their adventures, Bilbo had obviously earned his 1/14th share, so Thorin fulfilled his promise and sent Bilbo home with a chest of treasure (though it is hinted later on in LotR that Bilbo ended up receiving more).