First of all, the play is set through the Restoration period, when England came back to a monarchy 11 years after the ruling of Oliver Cromwell. Comedies of Manners are staples of the literature of this period.
Second, they are meant to ridicule the behaviors and specific mannerisms of a social class, particularly those of the upper classes.
Third, there are characters with roles that are typical for a good plot: In Way of the World, you have the man in love, the woman in love, the bad lady plotter, the snobby man, the bad guy, etc. All are very "black and white" characters as if pieces in a game.
Lastly, a comedy of manners is based on one thing that is hard to get and, in the end, it is attained. In the case of Way of the World, this thing is love. The tricks and trips that the characters have to confront in order to achieve love are what this is all about.
More examples of comedies of manners are: Sense and Sensibility and The Importance of Being Earnest. If you notice there are many similarities between these two story lines and that of Way of the World.