Despite being set so long ago in an unfamiliar New Dutch world, what about the story is essentially American and appeals to American readers?

Washington Irving's "Rip Van Winkle" is an early work of American fiction, and it is probably so appealing to Americans because it gives an account of the transition from British rule to independence in the fledgling nation.
The early portion of the story gives readers a look into colonial America, particularly New York in the area of the Catskill Mountains and the Hudson River. Rip Van Winkle is descended from Dutch settlers, and those familiar with colonial history will know that New York was the home of many Dutch transplants in the seventeenth and eighteenth centuries. Rip Van Winkle is a rather idle man who likes to socialize, and he seems opposed to productive labor; he is also repeatedly chided by his wife, who is portrayed as a dominating nag. Rip Van Winkle's central character trait, his laziness, is what ends up allowing us to see how the colonial world transformed into the American world—at least from Rip's perspective—seemingly overnight.
One day, to escape his wife's nagging, Rip goes off to the mountains where he observes a mystical scene and then falls asleep. When he wakes up and returns to town, he is shocked to find it changed. He does not know anyone in the village and everyone is dressed in a strange way to which Rip is unaccustomed. Much to his joy, Rip finds that his wife has passed away. Perhaps most significantly, Rip observes some changes to the former village inn:

Instead of the great tree that used to shelter the quiet little Dutch inn of yore, there now was reared a tall, naked pole, with something on the top that looked like a red night-cap, and from it was fluttering a flag, on which was a singular assemblage of stars and stripes— all this was strange and incomprehensible. He recognized on the sign, however, the ruby face of King George, under which he had smoked so many a peaceful pipe; but even this was singularly metamorphosed. The red coat was changed for one of blue and buff, a sword was held in the hand instead of a sceptre, the head was decorated with a cocked hat, and underneath was painted in large characters, GENERAL WASHINGTON.

Rip notes the presence of a flag that he has never seen before. This, of course, is the first version of the American flag, identified by its "stars and stripes." He also observes that the image of King George that was formerly there has been changed: it is now George Washington, general of the colonial army. Instead of a "red coat," representing the British, the figure wears "blue and buff." Further, the general holds "a sword" rather than a king's "sceptre." The general wears a hat instead of a crown. This passage encapsulates the changes that have occurred while Rip has been asleep. The American colonists have declared their independence and are fighting to be free of British rule; in the beginning of the story, Rip was a colonist and subject of King George. Rip notes the bustling activity: he sees speeches, encouragement to vote, and other activity we might expect in the public square of a village undergoing serious political change. He is somewhat indifferent when he discovers that he is now independent and free from the British, but he is thrilled to learn that he is free from the rule of his wife. He also is happy to discover that he is now an old man, so it is acceptable to no longer be productive. Even though he has done nothing to earn or deserve it, Rip has gotten everything he wanted in the beginning of the story.
The early American story "Rip Van Winkle" gives us a look at colonial life in New York before the Revolution, as well as a sense of how drastic the change was from colonial life to becoming part of a new nation. Because of that connection to American history, told through an entertaning and enjoyable story, "Rip Van Winkle" continues to educate and delight American readers.