Why do the King and Duke give the money to the girls in The Adventures of Huckleberry Finn?

The fraudulent king and duke are pretending to be the deceased Peter Wilks's brothers and the true heirs to his sizeable fortune. In this way they aim to to cheat Wilks's three daughters, Mary Jane, Susan and Joanna, out of their rightful inheritance. However, they give the money to the girls at first, because at this stage they're making a great show of caring about them and getting everyone on their side, and they plan to steal the money back later anyway. They certainly manage to fool most of the townspeople for a time - and in fact the girls are so taken in by their show of sorrow and sentimentality over Peter's death that Mary Jane actually hands the money back to them, saying that she trusts them completely. Huck, who is forced to go along with the king and duke, is absolutely disgusted with the whole thing, and resolves to reveal the truth to the girls. 

Why do the King and Duke give "their" inheritance to the girls in "The Adventures of Huckleberry Finn"?

They want to appear generous and honest so that they can gain the family's trust. It seems that the King assumes that he and the Duke will steal the money back later, but when I read this section I assumed the idea was to find a way to manipulate the family into giving the money back to them, which is what eventually happens. 

I agree with those who say that a con artist will first try to ensnare victims by winning their trust.  Huck knows that they are con artists, but he feels helpless.  He tries to think of a way to help without directly interfering with the king and the duke.

You must remember that the King and the Duke are con men! Every move they make serves only to further their OWN intersts. They are not really giving it to the girls; they just want it to LOOK like they are. Remember what is happening around the King and the Duke when they decided to give the money away. The town Doctor is questioning their identity and is trying to sway the townspeople to turn against them. At this critical moment where their cover may blown, the two con men decided to "give" them money to the girls to show everyone how "real" they are. The doctor is not fooled;however, the rest of the townspeople again begin to support the King and the Duke--their ruse can continue!

They want to appear gracious, selfless, and wonderfully supportive, in order to clamp down on any suspicions about their identity.  Besides, they aren't really handing the money over to the girls physically; they are still hiding the gold, and they can just say they are going to give it to them.  That sounds awfully nice, and no harm is done, right?  It helps to win the trust of the family and locals, and weighs in their favor when they decide to auction the house and property.  So, "giving" the money to the girls is just a cajoling tactic, a sweet gift to lure the family into trusting them, and to aid in further and more insidious plans.

Why do the King and Duke return the inheritance in "The Adventures of Huck Finn"?

The Duke and the King's motivation has nothing to do with guilt or any type of noble thought.  Because some are starting to question their identity as the heirs, they need to perform some gesture that will win unconditional trust from the Wilkes sisters.  By giving back the inheritance, they demonstrate their interest in the ladies' well-being and seem to be nobly motivated.  They have "studied" the daughters enough to know that they will not keep the inheritance, but rather that they will feel that they can trust the King and Duke with the care of the money. Thus, the naive girls give the money back to the con artists and they end up with more money and property under their control than before they "graciously" give back the money.