It is an archaic term from the early days of open-boat whaling, when the animals were harpooned from small open boats.
The term refers to Nantucket, Massachusetts, the center of the American whaling industry; as well as the speed associated with riding in a horse-drawn sleigh.
The sperm whale caused the longest drag events, reaching speeds of 23 mph (37 km/h).
Once the whale had exhausted its energy, the sailors pulled in the rope, killed the animal, and rowed it back to the whaling ship, where they would butcher the carcass, remove its blubber, and render its oil.
Sailors no longer needed to risk their lives in whaleboats, and so the Nantucket sleighride came to an end.