The Chapter you want to look at is Chapter 8 for the answer to this question. Let us remember that before the battle with the spiders Bilbo Baggins hadn't really done that much to help the group - he had been perhaps more of a hindrance than a help in their eyes. Therefore, what he is able to do in terms of freeing all of the dwarves single-handedly and the way that he is able to show himself a valiant warrior in fighting the spiders immediately changes their impressions of him. Chapter 8 says:
These questions they asked over and over again, and it was from little Bilbo they seemed to expect to get the answers. From which you can see that they had changed their opinion of Mr. Baggins very much, and had begun to have a great respect for him (as Gandalf had said they would)... They knew only too well that they would soon all have been dead, if it had not been for the hobbit; and they thanked him many times. Some of them even got up and bowed right to the ground before him, though they fell over with the effort, and could not get on their legs again for some time.
The text goes on to say that even though they knew about the existence of his magic ring, this did not diminish the new-found respect they had for Bilbo, because his exploits had shown that he has wits as well as "luck and a magic ring."