Pirates has always been a hit. Back when Walt Disney World opened, guests demanded the attraction be built as there was one in Disneyland.
No, no *defined* story. But as MKCustodial says, that's part of the magic of it. This can be paralleled to such classics as "It's a Small World" and "Journey into Imagination."
However, the Imagineers did create a story. A story based on history, myth, and mirth. It's our job to discover that the bones we see at the beginning are those of the pirates we will 'go back in time' to discover. As we meet up with the mayor in the well, the auction, the pirate in the pigs, the pirate trying to steal goods to his small boat, and the jailed pirates, enchantment reigns. A sensational feeling of nostalgia arises for Walt Disney World and Disneyland fans through the "touch" inspired by Walt Disney, similarly to those of the other attractions he never got to see completed: Country Bear Jamboree, Hall of Presidents, and The Haunted Mansion.