The choice of narrator/narrative style has a tremendous impact on the story. Because we learn things from Huck's perspective and as he realizes them, we see how and why they impact Huck.
Since the story is meant to showcase the change in Huck's character as he travels with Jim down the river, this choice of narrator is very important and effective. We see, for example, as he transforms from someone who refers to Jim as "Miss Watson's nigger Jim" (a clear indication that he buys into the mindset of his society) into someone who chooses to defy his society when he says "All right then! I'll go to hell!" and rips up the letter informing Jim's owner of his whereabouts.
Twain also chooses the first person narrator to allow him to include the use of the vernacular. Hearing the story told by an untutored teenager allows him to achieve the realism he is striving toward. It also allows for his satire to be that much more biting.