There are several reasons. First of all, they are grieving the passing of their friend, father, and neighbor. In this state of grief and confusion, two guys show up to relieve their suffering. They show up and are benevolent, kind, giving, and charming. It relieves their grieving a bit, and provides relief; it reminds them of their dear departed, and what a nice guy he was, and how lucky he was to have relatives that care so much. It's an easy lie to believe in the state of grief that they were in. Plus, the duke and king were acting as protectors, offering the girls a new life; without that, the girls are just alone in the world, fending for themselves. It was an easy lie to believe, since the alternative was frightening to them.
Secondly, what evidence do they have that they aren't the Wilks brothers? No one has ever met them; they come from England, and they seem to know all of the details of their lives, so why not believe them? How would they disprove it? And, the brothers are so kind that they wouldn't even be suspicious. They give the money back, offer to take the girls to England, and seem genuinely concerned for the family, even if they are a bit unorthodox. Because no one has any way of disproving them, they don't even try.
Thirdly, what the duke and king is so far out of the realm of expectations that I doubt most of the people there even thought to question them about their identities. Who would ever expect imposters to come in claiming such a huge lie? No one expects that; no one even wonders (besides the rather inquisitive Joanna who drills Huck about his origins, and Dr. Robinson; they are both quickly hushed though and that is the end of that).
Those are just a few reasons; I hope that helps!