The poem sketches a wealthy man at his club and doesn't have much kind to say about him. The opening simile, "private as a fart," is crude on purpose. It deflates any dignity the man might claim, suggesting his secrecy is less a mark of refinement than something faintly embarrassing, a thing everyone politely pretends not to notice.
The rest fills in the picture. He keeps his wit "in an after-dinner glass," which hints that his cleverness depends on drink, or that he only lets it out in comfortable, well-fed company. Then comes the sharpest turn: he keeps his family "in a false / laborious account." That could point to money managed dishonestly, or to a story about his household that takes constant effort to maintain. The poem doesn't say which, and the ambiguity is part of its sting.
The speaker seems dry and sardonic, watching from a distance with a raised eyebrow. The mood is cool, brief, and quietly contemptuous, like a sneer delivered under the breath.