One reason is that the Duke and the King did their research before going into the scam. They didn't just show up and claim to be the Wilks brothers, they learned the history of the family, town, and everything else that the Wilks brothers would be expected to know. They get the guy who let them know about the Wilks brothers totally drunk, and milk him for all of the information they need. When they show up knowing all of the names of the families, pets and neighbors, and asking about personal experiences and trials they've had, who's not to believe them? How else could they know these things?
The second reason that they were so easily duped is because of their ignorance. Twain loved to make fun of the ignorance of people in the south. These people haven't travelled, and aren't very educated; so, they haven't heard a real English accents, and don't know about English towns and places. So, when the Duke and the King show up, speaking in an English accent, dropping fancy, English-sounding names, they believe them because they don't know better.
A last technique that the Duke and the King use is kindness. They offer to give away all of the money and to take care of the family for the rest of their lives. That kills any doubts that people might have had. Why would impostors give away the profit from the scam? They kill doubt with excessive "generosity" and kindness, feigning that they don't care about the money. This makes people like and believe them even more.
I hope that those thoughts helped; good luck!
In my opinion, this is because all of these people are rural people who are essentially not very worldly. They do not know very much about what things are like in the larger world and so they are very easily fooled.
The more "backwoods" areas of the US in those days were really quite cut off from the more sophisticated areas. There was no TV or internet to let rural people see more of the world. Therefore, it was easy for them to be taken in by people who came in seeming to be more knowledgeable and worldly. The townspeople would have been a bit intimidated and would have been unlikely to challenge the duke and king who seemed much more knowledgable.
Because they pretend to act british using a fake accent