Satire is a genre of literature in which certain things are exaggerated, ridiculed, mocked, or criticized. This can be done in the form of a parody, exaggeration (hyperbole), metaphorically, and so on. In this story, "A Story," the narrator is a young boy commenting upon his uncle's annual excursion: essentially a day-long bar crawl. First, he satirizes his uncle and aunt. His uncle is described as a large buffalo, too big for his own home. His aunt is like a mouse, so quiet that she is hardly noticeable. The narrator satirizes the notion that his uncle is (or behaves as) this large, powerful, hard-drinking beast while his aunt is small and ineffectual. This is a satirizing of traditional (and typically Welsh?) male/female roles. The narrator is not overtly challenging these roles, but he does exaggerate the aunt and uncle in this way. 

The narrator also describes the bar (public house) crawl in fantastic terms, as if it were an epic journey, when in fact, it is only a bar crawl: 

And dusk came down warm and gentle on thirty wild, wet, pickled, splashing men without a care in the world at the end of the world in the West of Wales. 

The men are simply using the outing as a chance to escape from the daily boredom of their lives. They embrace the outing as if it is an escape into another world; they go to "the end of the world." This is part of the appeal of the event. It is a Romantic (in the sense of imagination) escape to an alternate reality. The men engage in witty banter and inside jokes as if they're speaking another dialect or language. The satire exaggerates this "epic" outing partly in making fun of its "epic" appeal but also in a non-critical, almost celebratory way.