How did Gandalf get Bilbo to agree to join the dwarves?

The response was: Fili and Kili looked uncomfortable and stood on One leg, but the others made no pretence of offering - except old Balin. the look-out man, who was rather fond The Hobbit. He said he would come inside at least and perhaps a bit of the way too, really to call for help if necessary.
Fili and Kili were embarrassed but did not offer to go. The rest of the group, save Balin, did not even make a pretense of embarrassment and were fine with Bilbo going in alone. Balin offered to go inside a little way and call for help if needed.