Ironically, in "The Storyteller," it is the inner satiric story that exemplifies more about the art of storytelling than its frame story, while providing more insights about human nature as the children recognize "a ring of truth" in the bachelor's story and pronounce its ending as "beautiful."
The children enjoy the bachelor's story because it breaks from the stodgy. Having been raised by a rigid aunt himself, Saki ridicules the oppressiveness of the social ritual of preaching good behavior and social graces by having his bachelor storyteller break from traditional narratives in appealing to the sadistic innateness of the children. For, it is the very goodness--"horribly good"-- of the little girl that causes her demise as her medals for good conduct, obedience, and punctuality clink against each other as she hides from the wolf, drawing his attention back to her as his prey.
In addition, Saki has his storyteller triumph over the mundaneness and respectable self-interest of the aunt through the power of his imagination. When, for instance, the bachelor tells the children there were no sheep in the park of the Prince, he seems to have failed with the children and the "aunt permitted herself a smile...almost...a grin." However, he quickly recovers with a deft imagination:
"There were no sheep...because the Prince's mother had once had a dream that her son would either be killed by a sheep or else by a clock falling on him...."
The aunt suppressed a gasp of admiration.
With his imaginative story, then, the bachelor satirizes convention as he proves that what is morally instructive is not always interesting to children.