Epithets are similar to nicknames; they are alternate titles for something or someone which describe some specific quality or characteristic. Bilbo uses epithets to describe himself because he doesn't want to give away his real name.
Bilbo is sent to investigate Smaug's lair within the mountain for several reasons, most specifically because it was the job that he was initially "hired" for at the beginning of the journey. Smaug, having a well-earned reputation for being deadly, not to mention having been out of sight and up to who-knows-what type of evil in the intervening years, inspires Bilbo to wear the Ring before approaching him. This pays off, because Smaug can smell him and feel the air that he displaces while moving, and so most of their conversation is tinged with Smaug's anger and subtle attempts to trick or bully the hobbit into revealing himself.
In one of the book's many expositions, which have a tone of confidential just-so aphorisms, it is said that refusing to answer a dragon is unwise, and it is equally unwise to tell him exactly what he wants, specifically your real name. Thus Bilbo gives several names for himself which describe his personality traits, personal history, and accomplishments; for example, "Luckwearer", "Barrel-rider", "web-cutter", "chosen for the lucky number" and "he that walks unseen".
Bilbo's use of epithets at this time also reflect a degree of character growth; at the beginning of the journey, Bilbo was reluctant to be considered a burglar, or to find much enjoyment in the job. However, as he grows more confident and his skills become more evident, he begins to take pride in himself and his accomplishments and become more proactive and self-advocating. His use of epithets, and doing so in a riddling manner like that which he used with Gollum, are a sort of self-reflection of the many accomplishments Bilbo has achieved in his journey thus far.