This quote is taken from Chapter 9, which is when Bilbo is wandering around the Elvenhalls by himself with the ring on whilst the dwarves are captured by the King of the Elves and imprisoned. Bilbo is forced to stay within the Elvenhalls and to steal food in order to survive, as he thinks desperately about what his next move should be. He is clearly very unhappy at this point in the narrative, but note what the narrator tells the reader just after the quote cited above:
He often wished, too, that he could get a message for help sent to the wizard, but that of course was quite impossible; and he soon realised that if anything was to be done, it would have to be done by Mr. Baggins, alone and unaided.
The significance of the quote therefore lies in the way it presents a key feature of Bilbo's identity and also his strengths as a character. Gandalf suggested him to the dwarves precisely because he is a talented burglar, and the quote cited above indicates the way in which Bilbo is more and more identifying himself in that role that was formerly only given to him by Gandalf. It also represents his increasing character development, as he sees that if any resolution to this situation is going to emerge, it will have to come from him.