The prologue, written by the famous actor David Garrick, suggests that the comedic arts are dying, and that, if that should happen, comic actors like himself will lose their livelihoods. He suggests that Goldsmith might be a "doctor" that can save comedy and that his "five draughts" (the five acts of his play, She Stoops to Conquer) might be the medicine needed to affect a cure, or in other words, to make the audience laugh. In fact, the prologue ends with a guarantee of sorts: if Goldsmith's play is funny, the audience will "give him his degree," or assert that the play is good— otherwise "he [Goldsmith] will receive no fee!"

The prologue was written in part to explain to the audience what they were about to see: this will not be a play about morality or manners but instead is meant to make the audience laugh. This distinguishes Goldsmith's play from the sentimental comedies that had come to dominate English theatre for much of the 18th century (one example is Steele's The Conscious Lovers). Those plays emphasized propriety and social morality; Goldsmith's play, which lampoons the "manners" of the upper classes, is about being funny.

David Garrick, the writer of this prologue, was one (if not THE) most famous actor and producer of his time. He wrote this prologue as a satire, where the character "Mr. Woodward" would be in  mourning, "because comedy is dead", and so they are hoping that Goldsmith's play would make him laugh again. Point in case where he says:

Excuse me, sirs, I pray—I can't yet speak—
     I'm crying now—and have been all the week.
     "'Tis not alone this mourning suit," good masters:
     "I've that within"—for which there are no plasters!
     Pray, would you know the reason why I'm crying?
     The Comic Muse, long sick, is now a-dying!

Another explanation to the prologue is that, in those days, comedies were not meant to make you laugh, but simply to tell a story with a happy ending. This prologue basically shows that this will be a "first" in theatre, and that times are about to change.

The link provided gives you the prologue in Modern English so you can understand it better.

http://www.ourcivilisation.com/smartboard/shop/goldsmth/stoops/prologue.htm