What is the conflict that the framework narrator faces, and how is this conflict resolved?
What is interesting about your question is the use of the word "conflict." I can tell by the way you are asking the question that you mean conflict as the "inciting incident" of the story, the one that needs a "resolution." This shows that you are referring to plot, ... and not a conflict that sets one person vs. another, or a frog vs. a person, etc. So let's discuss conflict as the inciting incident of the plot with regard to the full plot. (Disclaimer: Please realize that I will be discussing the plot IN THE CONTEXT that this is a frame story! The frame of the story is simple exposition and denouement.)
The exposition of the story includes the frame and learning about the gambling issues of Jim Smiley. The inciting incident, the conflict, in the plot is Jim Smiley making a bet with his "celebrated jumping frog." The rising action creates suspense for the reader when the challenger fills the frog with buckshot (little metal balls) in order to keep the frog from jumping. This is done without Jim Smiley's noticing. As a result, the climax of the story is that the frog doesn't jump. Here is the climax:
Then he says, “One, two, three, jump!” and him and the feller touched up the frogs from behind, and the new frog hopped off, but Dan'l give a heave, and hysted up his shoulders so like a Frenchman, but it warn't no use he couldn't budge; he was planted as solid as an anvil, and he couldn't no more stir than if he was anchored out. Smiley was a good deal surprised, and he was disgusted too, but he didn't have no idea what the matter was, of course.
If this is the climax, then the resolution is that Jim Smiley, the famous gambler, loses the bet. (The denouement, then would be the end of the frame story.)
In conclusion, it's important to note that this is a frame story (in that the narrator was sent to learn something and does, but it is only in this regard that we learn about the frog). The frame really has nothing to do with the inciting incident of the plot. In this regard, and as you can see, I disagree with the first two responses.
To me, the major conflict that the narrator faces is how to get out of the situation he finds himself in. He has gone to look for Leonidas Smiley and is now having to listen to all these stories about Jim Smiley. He is bored stiff and wants to get away.
What he ends up doing is just leaving. Simon Wheeler hears his name called and gets up to see what is going on. At that point, the narrator makes a hasty escape.
This is supposed to symbolize a tension between the cultured people of the East and the supposed hicks of the West.
The conflict is that he is sent to learn about the Reverand Leonidas W. Smiley and Simon Wheeler proceeds to tell him about Jim Smiley. A trick has been played on the narrator which he discovers soon after Wheeler begins speaking. The conflict is resolved much later after Wheeler is called away and the narrator is able to escape without being further tortured by stories from this very talkative gentleman.