In chapter 10, why does the Master decide to help the dwarves continue their quest to recover their treasure?
Chapter 10 of The Hobbit is titled "A Warm Welcome." It comes at the end of the entire Mirkwood forest sequence where the company of dwarves and Bilbo have dealt with giant spiders and being captured as prisoners by the woodland elves. Bilbo devised a plan to sneak himself and the dwarves out of the Elvenking's fortress by hiding in barrels and floating down the river and out of the forest altogether. The company floats down the river in the barrels until they come to the Long Lake which is home to Lake Town, a community built on the lake that is inhabited by Men (humans).
Lake Town is governed by the Master, a man who can be considered the Middle Earth version of the mayor of a town. The dwarves (lead by Thorin Oakenshield) are on a mission to reclaim their ancestral land (and the considerable treasure it houses) of the lost dwarvish kingdom of Erebor inside what is called "The Lonely Mountain." Thorin is the grandson of Thror, the final king of the mountain when the dragon Smaug claimed the mountain and its treasure and scattered the dwarves for good. The people of Lake Town are very excited when they see the dwarves arrive because the Men of the town remember the old songs that speak of the dwarves reclaiming their kingdom and
... gold would flow in rivers, through the moutain-gates, and all the land would be filled with new song and new laughter.
The Master, however, is a very practical man, and he is not entirely convinced of the stories contained in these old songs and legends, nor is he convinced that Thorin is really a descendant of the line of dwarf kings of old. And he certainly isn't convinced that this ragged crew could ever defeat a dragon. But he sees that the people of the town are very excited over the arrival of the visitors and he allows the company to rest and rejuvenate for a couple of weeks, eating and drinking very well and participating in songs and merriment. But ever the practical man, the Master is ultimately glad to get them out of town and on their way.
But the Master was not sorry at all to let them go. They were expensive to keep, and their arrival had turned things into a long holiday in which business was at a standstill.
So the Master gives the dwarves provisions and transportation up the river and helps them on their way not so much with the expectation of eventual repayment in the form gold. He mostly just wants to get rid of them to be rid of their expense and to allow the commerce of the town to return to normal.