The "framing" of this story through a narrator (Crayon), and two authors (Knickerbocker and Irving, who are likely one and the same) sets a mock-serious tone from the very beginning. It would seem that the purpose behind such framing is to add to the credibility and truth of the story. In reality, the introductory passages are very "tongue in cheek" (or satirical).
What this framing actually does is emphasize the story's nature as folklore and fiction and add to its humor. The story itself (without the quote and note of explanation at the beginning as well as the extra information in the afterward) sounds pretty silly. Think about it. A lazy man goes off into the woods to escape his nagging wife, gets drunk with a bunch of forest-bowling Dutchmen, and wakes up 20 years later to find that his nagging wife is dead and his own (now grown) daughter will allow him to continue in his laziness and be respected for it?! How is this plausible?
By framing the story behind a narrator who claims he found the story among the papers of the revered "Knickerbocker" is an overt display of further silliness and humor. It is almost as if the humor of the story itself is not enough. It must be heightened by a story teller who, if you could see him, would undoubtedly possess that familiar twinkle in his eye and slight curve in the corner of his mouth that such story-tellers possess when they think they are pulling one over on their audience, but deep down wonder themselves if they are actually getting away with it.
The result of the framing is that a seed of doubt is planted. The beauty of the story that follows, then, is the wonder of whether it is true or not, in the face of so many factors that suggest it is, but so many details that confirm it cannot be.
The italicized opening of Washington Irving's "Rip van Winkle" is more than mere introduction. It provides a frame, supposedly explaining how the tale was found, and defending its credibility. But, unlike the "Custom House" which provides a most serious and intriguing frame to Nathaniel Hawthorne's The Scarlet Letter, Irving's frame adds mostly humor to his story since Knickerbocker is clearly an eccentric character of Irving's imagination.
Aspects of Romanticism are, for instance, Irving's dwarfing of Rip's domestic problems by the beauty and majesty of nature. Lost in the Katskills, Rip van Winkle is calmer and contemplative, much like the Romantic. Certainly, Irving's own Romantic ideals are expressed in his satire of his contemporary post-revolutionary society, which he suggests is too argumentative, rationalistic, and dogmatic:
In place of these a lean bilious-looking fellow with his pockets full of handbills, was haranguing vehemently about rights of citizens--elections--members of Congress--liberty--Bunker's hill--heroes of Seventy-six--and other words which were a perfect Babylonish jargon to the bewildered Van Winkle.
When Rip arrives, the bystanders shout "A Tory!...a spy! a Refugee!" Rip inquires about his old friends, only to be told that they have died or gone off.
Rip's heart died away at hearing of these sad changes in his home and friends, and finding himself thus alone in the world--every anwser puzzled him too by treating... of time and of matters which he could not understand....
As a Romantic, Irving expresses nostalgia for the stability, calm, and natural beauty of the colonial village.
In my opinion, Irving's tone in the introductory passages is sort of ironic and amused. It seems to be that he is looking at Knickerbocker's work in a patronizing way. I do not think he really "believes" it. I say the tone is amused and ironic because he sometimes says that the stuff is absolutely 100% guaranteed true, but then he says that Knickerbocker is sort of strange, etc.
Just after the introductory stuff, you can see examples of romanticism. Look at how beautifully he describes the mountains. The description makes me want to go and have a look and is clearly written by someone who loves nature. For example:
When the weather is fair and settled, they are clothed in blue and purple, and print their bold outlines on the clear evening sky; but sometimes, when the rest of the landscape is cloudless, they will gather a hood of gray vapors about their summits, which, in the last rays of the setting sun, will glow and light up like a crown of glory.