The offstage activities in The Alchemist make it easier for gullible characters to be duped and for the plot to unfold along comic lines. (Much of the humor comes from dramatic irony—the audience knowing what many of the characters in the play do not.)
First, it is Lovewit fleeing his house to live "offstage" because of the plague that allows the lowlife con artists Subtle, Face, and Dol to take center stage as they use his home to try to cheat others.
Subtle's supposed work on alchemy—turning baser metals such as lead into far more valuable metals such as gold—takes place offstage, allowing the con artists to string Mammon along as they take his money in return for allegedly eventually giving him the secret to this alchemical process. Subtle is, of course, not working on real alchemy at all. However, in act 4, scene 5, he does manage to create a false explosion in his supposed lab. Because all his work has happened off stage, he is able to convince Mammon that the explosion was due to Mammon's lust—Mammon's lack of purity in being with Dol, Subtle says, ruined the alchemy.
The explanations for offstage events help dupe characters who are eager to believe that their dreams can come true, and who are willing to turn a blind eye into looking too deeply into what might ruin the fantasy. The play suggests that we should all look around the corner at what is behind the scenes if something appears too good to be true—but also suggests that it is human nature not to want to.