Beside cartoons and movies, why does "The Legend of Sleepy Hollow" continue to have meanings for readers today? What about the story is essentially American and appeals to American readers?

"The Legend of Sleepy Hollow" resonates as an American story today because it highlights the virtues of American heartiness and pragmatism—the frontier spirit—which win out over effete intellectual learning and the desire for unearned wealth associated with Europeans.
From the start, Ichabod Crane, the outsider, is described in terms that make him suspect. While one might think a newcomer to a town is more likely to embody the American spirit, Ichabod brings with him too much old-style baggage. He is described as,

A kind of idle gentleman-like personage, of vastly superior taste and accomplishments to the rough country swains.

We also learn that "country bumpkins . . . envy . . . his superior elegance and address."
Words like "gentleman," "superior taste," and "superior elegance" are not congruent with the self-made, equalitarian American spirit. Further, other descriptions of him foreshadow that he will not win out against a red-blooded American male like Brom Bones:

One might have mistaken him [Ichabod] for the genius of famine descending upon the earth, or some scarecrow eloped from a corn-field.

When Ichabod and Brom Bones become rivals for Kristina's hand, there is not much doubt that the ‘‘burly, roaring, roistering’’ Brom, the "hero of the country round," will win the day. Brom has a gang of male friends and followers, while Ichabod is only a favorite with the women. Ichabod is crafty, while Brom described as follows:

Always ready for either a fight or a frolic; but had more mischief than ill-will in his composition; and, with all his overbearing roughness, there was a strong dash of waggish good humor at bottom.

Ichabod seems to lust after Katrina's food and pewter more than Katrina herself. In a stereotypically European fashion, Ichabod wants to marry up to unearned wealth, while the bold Brom loves Katrina for herself.
In the end, Brom is able to play on Ichabod's overeducated fears, bred of too much reading abut "the direful omens and portentous sights and sounds in the air, which prevailed in the earlier times of Connecticut." Ichabod falls for the headless horseman stunt, and Brom wins the girl.
As in "Rip Van Winkle," energy, masculinity, and hardworking pragmatism represent the positive appeal of Americanism, and are contrasted with the more apathetic and lazy qualities of the Old World represented by Rip and Ichabod. To this day, "know-how" and the qualities of rugged individualism and resourcefulness tend to be associated much more with the American spirit than intellectual knowledge and "superior elegance."