Hello Stan.
First, please realize that tutors here on Wyzant are, most likely, not going to simply tell you, "oh the tone is buoyant" or "the mood is caustic" or "the mood of the story is flippant". (To be clear, I don't think the tone of this story is any of those words)
Next step: could you clarify the question you are being asked for the assignment? "Tone" sometimes means, "the author's attitude toward a topic/subject". "Mood" can mean something slightly different -- it's the "feel" or "atmosphere" of a story. But some teachers/professors will use "tone" and "mood" interchangeably, and, if so, the assignment is probably asking for the mood/attitude/feel/atmosphere of the story. That's why I asked for clarification of the assignment.
That being said: you're going to need to find a strong, specific adjective to describe the tone/mood.
Step 1: I would suggest googling "some words to describe mood of a story" just so you have a bank of specific words at your disposal.
Step 2: Re-read the story. The short version: the bachelor on the train is so annoyed by the children on the train and by the aunt's inability to entertain them that he tries to prove that he can do so himself. He tells a seemingly pleasant story which ends rather gruesomely. The children love the story. The aunt isn't so happy about it. The bachelor is pleased because he knows the kids will later ask for more of these types of stories.
Step 3: Let's assume your teacher is asking about mood. If "The Storyteller" were a ghost story, you might describe the tone as "creepy" or "tense" or "ghoulish". Or if it were a love story, you might describe the story as "romantic". But it is neither of those things.
So one tip I would suggest is to consider: what would Saki (who died in 1916 I think) tell you the purpose of this story is, if you were sitting around having a cup of tea with him? Was he trying to amuse the reader? Was he making fun of any character in the story? Was he trying to present any of the characters as a hero? Was he trying to teach a lesson through this story?
Step 4: Please consider the questions I've posed here, and post your initial thoughts on the story and those questions here in the chat. I'm sure one of the tutors will give you some feedback on what you post.