Great question! I can't really speak to their ubiquity in modern pop culture, but here are a few reasons why the pirates of the "Golden Age" of piracy were so popular in their time:

1. Frustration with social structure. Social mobility in the 17th and 18th centuries was nearly impossible. As such, members of the lower classes had a certain reverence for their fellow countrymen who were willing to disregard the class hierarchy and take control of their own destinies by any means necessary. Following this general sentiment, folk stories such as Robin Hood were very popular in their day. Some pirates, Samuel Bellamy in particular, regarded themselves as 'Robin Hoods of the Sea,' thus endearing them to commoners who dreamed of having a better life no matter the cost.
2. Anti-foreign sentiment. Mid to late 17th century pirates (e.g., Henry Every) committed what could only be described in modern times as unspeakable atrocities against foreigners in the Indian Ocean. The accounts of these atrocities were re-cast for English audiences to portray the pirates as essentially heroes fighting against non-Christian heathens. They even wrote plays/stories/etc. that glorified what would otherwise be considered as disgusting crimes.
3. Politico-religious conflict. Following the end of the Stuart Dynasty in England, there remained a large contingent (especially Catholic but also non Church of England Protestants) who, for various reasons that I won't elaborate on here because they can get complicated, felt that the deposition of King James and the Stuart lineage was unlawful. They fought to restore King James's lineage to the throne, participating in attempted coups on several occasions, but were ultimately defeated. Many of these supporters of King James, 'Jacobites,' ended up fleeing England. Some went to Ireland, France, etc. where their beliefs were supported. Others, as indicated by artifacts and testimonies (see 'Real Pirates' by Barry Clifford and Ken Kinkor, 'Pirates in their Own Words' by E.T. Fox) apparently joined up with the pirates. Some historians speculate that their ultimate goal was to wage war against the English Protestant government via piracy, funneling proceeds into the military-backed restoration of King James's lineage. If you are familiar with the American Civil War, I equate this to the (Jessie) James Gang; though Jessie James and his gang are often portrayed as simple, money-hungry outlaws, in reality they refused to accept the surrender of the Confederacy and wanted to mount a grass-roots effort to help the 'South rise again' via robbery.
4. Pop culture. In terms of news/excitement/gossip/etc, the 17th and 18th centuries could be considered slow and boring by modern conceptions. News took weeks/months/years to travel and was not readily accessible for many. However, with the growing availability of printed media in this time period, mostly newspapers but also novels (e.g, Robinson Crusoe, the 'first English novel,' by Daniel Defoe, 1719, largely inspired by piracy and exploration), anything 'saucy' or controversial was devoured openly by the common man, and, despite their public prudishness, by upper classes in private. To cast this in modern terms, think about the popularity of shows/movies about criminals; Narcos, The Sopranos, Cocaine Cowboys, Good Fellas, The Godfather, etc. We all love stories about criminals, and even if we completely disagree with criminal behaviors on a moral level, many of us secretly dream about what it would be like to be in a position of power like that. People in the 17th/18th centuries were no different.
5. Psychological innovation. This particular comment speaks to your question about the Caribbean pirates. Prior to the golden age of piracy, buccaneers, privateers, etc. generally operated through what might be described as terrorism; brute force, lots of gunfire, no quarter, etc. While brutal, this was also typical for the day (and probably for thousands of years before). The 'Flying Gang,' which consisted of mostly English, but also French, pirates operating out of the Caribbean in the early 18th century, figured out that psychological terror was a far better weapon than brute force. Whereas previous privateers/pirates raised plain flags (red or black) to indicate 'no quarter,' the Flying Gang started using flags with scary images, e.g., skulls, devils, weapons, blood, hourglasses to scare the hell out of anyone they were chasing. If some sources are to be trusted, they also started cultivating fearsome images (esp. Blackbeard!), not because they were actually maniacs but rather because they understood that anyone who believed they were real devils would be more likely to surrender without a fight. Presumably, all of this 'scary' stuff further captured the attention of the public and created the modern pirate mythologies that we are familiar with.

I hope that helps! There is obviously lots more to discuss in regards to your question. I would suggest checking out books by E.T. Fox, Richard Zacks, Jan Rogozinski, and Colin Woodard.

The first reply to this is great, and though I’m answering very late(I’ve been getting into Piracy lately), I think another big thing to mention are the roaming grounds of the pirates during the golden age. The two biggest pirate strongholds in this era from 1650-1730 were the West Indies/Florida and West African Coast. Both places have beautiful water and strange but wonderful looking plant and animal life on land and sea. These places are far more exotic than Europe and they no doubt contribute to the popularity of pirates. A major appeal of the Pirates of the Carribean movies are the places the crew travels to. Port Royal in the first movie looks gorgeous. Much like how Cowboys are very popular in pop culture as well, the Caribbean was sparsely settled until well into the 19th century. The lawless feel of the Golden Age of Piracy makes it feel like a western on water to many readers

Similarly, on a visual level, I think the Golden Age being near the high point of the Age of Sail contributes a great deal. Many people find these kinds of ships to be stunning, fearsome and elegant at the same time. They perfectly compliment the roaving grounds that the pirates occupied.