According to the opening paragraph, the narrator's friend simply wanted to have some contact with his friend's friend Leonidas W. Smiley. The narrator's friend wrote him "from the East." In those days communication between the American East and West would have been slow and uncertain. The narrator's friend says in his letter that he thinks Simon Wheeler might know something about the whereabouts of Smiley. But the narrator says he suspects that his friend in the East was playing a practical joke on him. The friend wanted the narrator to meet Wheeler, who was a terrible bore and would never stop talking about Smiley and anything else that came into his mind. When the narrator finds Wheeler, he hears the story about Smiley's jumping frog--but he never does find out anything about the whereabouts of Leonidas W. Smiley.
I told him a friend of mine had commissioned me to make some inquiries about a cherished companion of his boyhood named Leonidas W. Smiley—Rev. Leonidas W. Smiley, a young minister of the Gospel, who he had heard was at one time a resident of Angel's Camp. I added that, if Mr. Wheeler could tell me any thing about this Rev. Leonidas W. Smiley, I would feel under many obligations to him.
This succinctly explains the narrator's original purpose, but, as he later reflects, he thinks his friend in the East was playing a practical joke on him by getting him involved with an old man who would bore him "nearly to death." The story of the jumping frog not only characterizes Leonidas W. Smiley--or at least somebody whose last name was Smiley--but it also characterizes the garrulous Simon Wheeler. The most interesting aspect of Mark Twain's story is his recreation of Simon Wheeler's colorful use of the English language. Wheeler shows his garrulous nature and his quaint speech as soon as he starts talking:
There was a feller here once by the name of Jim Smiley, in the winter of '49 or maybe it was the spring of '50 I don't recollect exactly, somehow, though what makes me think it was one or the other is because I remember the big flume warn't finished when he first came to the camp; but anyway, he was the curiosest man about always betting on any thing that turned up you ever see, if he could get any body to bet on the other side; and if he couldn't, he'd change sides