There are two epilogues commonly published with the play. The second, intended to be spoken by Tony Lumpkin, was not written in time for the original production.
The first is intended to be spoken by Kate. The character begins by summarizing that she has "stooped to conquer with success" and that the author has thereby conquered his audience. She proposes that "our life is all a play" and then traces the five act life of a pretty country barmaid. In her first act, she is simple, afraid and eager to please. In her second act, she is loud and authoritative. She next moves to town where she impresses everyone with her character and charm. Her fourth act has her wedded to a man of repute, and pretending towards snobbish taste, she ends up losing her edge. Goldsmith (as author) is responsible for her fifth act, in which she might again become judge.
The second epilogue is attributed to J. Cradock, an actor and dramatist of the time. It is meant to be spoken in Tony Lumpkin's voice. Tony notes that, now that the play is done, the audience must want to know what happened to him. He tells how he will "in the great world appear," bringing his lively spirit to London where he will show the world what good taste is.
The second epilogue is cute and would likely be a fun transition out of the play, but does little to significantly further the play's theme. It certainly can be used as argument for the centrality of Tony to the themes, but at best, it offers the audience a reminder that “good taste” should come from a spirit of liveliness and not moral sanctimony or given assumptions about proper breeding and education, since Tony in most ways lacks those two qualities.
Contrastingly, the first epilogue is a nice summation of the goal Goldsmith set out for himself in his "Essay on the Theatre." Though it's not explicitly stated, the barmaid whose life he describes is likely meant to represent the theatre itself. She learned to confront her audience and demand things of them, then was brought to high society, where she grew pretentious and lost her edge (regressing into "sentimental comedy"), and now sits docile, waiting for someone like Goldsmith ("the doctor" from the prologue) to see where he can lead her. He wants to recapture her bawdy charm from her younger days, and he hopes he can "conquer" his audience by doing so. So the epilogue here serves as a challenge to the audience – did he succeed? Did he conquer them into accepting the low and bawdy nature of comedy again, leading them to repudiate their assumptions about high-minded theatre?