How does Twain capture the flavor of the old West in the story of "The Celebrated Jumping Frog of Calaveras County"?
I think it is clear that this story lives, breathes and eats the "Old West" that you refer to in your question. Twain was a master of capturing a particular moment and context in history so precisely that, in spite of the separation in years, you can really imagine the time and place he evokes. One central way that he was able to do this so successfully was through the way that he used the vernacular. The vernacular refers to the language commonly spoken by people in a particular place or religion. The use of the vernacular is the central aspect of Twain's style in so much of his fiction. Consider how its use helps create the characters and presents vividly the flavour of a mining camp in the gold rush era:
...he was the curiousest man about always betting on anything that turned up you ever see, if he could get anybody to bet on the other side; and if her couldn't, he'd change sides. Any way, that suited the other man would suit--any way just so's he got a bet, he was satisfied. But still he was lucky, uncommon lucky; he most always come out winner.
Such use of the vernacular greatly helps the reader imagine the characters and the setting, presented them in bold brush strokes that clearly bring to life a particular period of history and the characters from that context.