For Saki, satire is the tool with which he is most proficient; in addition satire serves Saki's purposes as an author.  For satire, while providing humor in its ridiculing of human foolishness or wrongdoing, ultimately holds a moral.  The sting of satire is meant to cure readers of their pretensions and blindnesses.  In contrast to the realistic, ironic writer such as O. Henry, Saki as the satirist wishes to reform the world in addition to providing it with some entertainment since the premise of a satirist is that by exposing an unacceptable situation to ridicule and laughter, this situation may not last.

His satiric stories about his Ewardian society, for instance, are meant to both ridicule and improve this society.  In his story, "The Storyteller," Saki clearly criticizes the aunt's permissiveness regarding the children under her care.

Unhappy woman!" he observed to himself as he walked down the platform of Templecombe station; "for the next six months or so those children will assail her in public with demands for an improper story!"

The storyteller hopes, then, that the aunt will examine her methods of child rearing and adjust a few things, punishing her for her remark,

"A most improper story to tell the young children!  You have undermined the effect of years of careful teaching."

Often Saki’s mischievous protagonists arrive on a scene to

"wreak havoc on victims who have invited their tormentors out of folly or a streak of viciousness of their own." [enotes]

just as the aunt has invited the man in the train compartment to tell his story.  So often, Saki's satire is directed in this way as, for example, the foolish Framton Nuttel arrives for Vera to "wreak havoc" upon in "The Open Window." 

I have included a few links below to various enotes study guide on Saki so that you can read more about Saki's life and his work. What is notable is the way in which his writing career began when he worked as a political satirist and foreign correspondent for a number of different newspapers. It was clearly his talent for satire that led him into writing, and this early work gained him a reputation for a very dry, ironic and satiric style that enabled him to gain an audience for his short stories that are so famous for their biting satire. In particular, Saki is famous for the way in which his fiction manages to poke gentle fun at the norms and values of Edwardian society. You might wish to analyse his famous short story "The Open Window" in this light, or "The Storyteller," each of which in their own way satirically presents Edwardian society in all of its glory and folly.