Why would a Revolutionary-era person read this story? Give specific examples.

The story is set in 1727 (the early eighteenth century), which is a significant time period in American history.
A Revolutionary-era person might be interested in Irving's story because it alludes to important philosophical, political, and religious struggles in seventeenth- through nineteenth-century America.
Here are three specific ones you may be aware of in the story:
1) The persecution of Anabaptists and Quakers by Puritans.
After his religious conversion, Tom Walker openly discusses reviving the persecution of Quakers and Anabaptists. In the seventeenth century, New England Puritans diverged from their English brethren in how they viewed religious matters. While English Puritans supported presbyterianism (church government through representative assemblies) and freedom of religion, their New England peers preferred a more independent system.
Yet, the New Englanders disapproved of Quaker beliefs, particularly the belief in female preachers, Inward Light (the belief that personal revelations are equal to or above Scripture), and orthopraxy. Orthopraxy differs from orthodoxy in the sense that it elevates behavior over belief and tradition. For more about the persecution of Quakers and Anabaptists, read Why The Puritans Persecuted Quakers and Quakers in America: From Persecution through Toleration to Domination.
2) The exploits of Captain Kidd, the infamous pirate.
This would have been the main element of interest for a Revolutionary-era reader. You can read about Captain Kidd's history here. In the seventeenth century, Kidd was a pirate of Scottish descent. His most daring exploit was to enlist King William's support in his schemes to plunder booty from pirate ships.
In fact, Kidd's actions at sea caused many English sea captains no end of consternation. They petitioned King William for relief, and eventually Kidd was apprehended. He was sent to prison in Boston, and at the time, rumor had it that he hid a large treasure underneath a thick grove on the mainland. This story is mentioned at the beginning of Irving's story.
3) The American Indian Wars.
The story mentions a fort from the days of the American-Indian wars. This fort is said to be the site of magic incantations and sacrifices to the devil. You can read about the American-Indian wars here and how Native Americans largely allied with the British against the Americans here.
So, mentions of events of religious and political import would certainly be of interest to readers from the Revolutionary era.