The very notion that there would be a young reverend in a mining camp is humorous because mining camps were notorious for all manner of immoral behavior and the absence of virtuous women and men of the clergy. The invented name of the absent reverend, "Smiley," is meant to emphasize the humor that the narrator has been sent on a fool's errand.
The Western dialect that Simon Wheeler uses is also meant to enhance the story's humor. Nonstandard words and phrases like "straddle bug," "foller' (for "follow"), "bannanner," (for banana), "bully-rag," and "sorter discouraged-like" would be funny, especially to a more educated, or perhaps Eastern, audience unused to such colorful language.
Smiley's description of the posture of the frog as it attempts to jump (it "hysted up his shoulders so like a Frenchman") is a humorous and harmless swipe at Europeans that would likely amuse the nineteenth-century audience.
The exaggeration of some elements of Wheeler's story also create humor. At the story's end, his claim that the frog "belched out a double handful of shot" is impossible and utterly ridiculous.