Act II is intended to be the most satirical and ironic act of the play. This is because Act II moves the plot forward and establishes the obstacles that will get in Jack's way as he tries to resolve the central problem of the play: that he needs to find a way to marry Gwendolen.
The obstacles that arise include falsifying identities, lying, pretending, twisting truths, and discovering secrets. These obstacles would render a very dramatic and somewhat tragic ending in a typical play. However, in a Wilde comedy, these factors are used to ridicule the serious and otherwise romantic nature of Jack's problem while accentuating Wilde's intention to make the play "trivial".
For these reasons, Wilde does his best at weaving into the dialogue as many of his epigrams and paradoxes as possible.
Well, I know, of course, how important it is not to keep a business engagement
is a prime example of the satirical and ironic way in which Wilde intends for his characters to show carelessness and oblivion about important issues.
When Algernon is told to leave the Worthing estate Cecily, who is besotted by him, does not really want him to leave and questions where he is going and what he is going to do.
Algernon, who is pretending to be Jack's "wicked" brother Ernest does not want to leave either, but explains his duty to do so. This is why Cecily says that it is very important NOT to keep an appointment. Making this point shows that Cecily is similar to Algernon in that she does not care for seriousness nor for doing the right thing at all times. The fact that Algernon fully agrees with her makes this evident.
Yet, Cecily is also the conduit that enables Wilde to add the contradictory statements that constitute the his signature comedic style. Therefore the use of the paradox breaks the courtship scene and divides it into romantic and silly at the same time.