That's a vast subject. Some time ago, a IAMA thread was opened on the subject by /u/DrRichardBlakemore here. There is a LOT of info about piracy and pirate, so if you are interested on the subject, I suggest you take a look.

 

To give a "quick" awnser to your question is difficult, because there is numerous factor in the rise and fall of the piracy in the Caribbean, but I can give you the basics. Most of my knowledge about pirates comes from Marcus Rediker's Villains of All Nations: Atlantic Pirates in the Golden Age, a book that I strongly recommand.

First of all, how did it came to be during the golden age of piracy, wich is, raughly, 1650 - 1730 ? The first explaination is the importance of the maritime trade routes in the Caribbean. The development of the plantations in the Caribbean and of the European colonies in Caribbean and American meant there was a lot of marchandise and money traveling around in the Caribbean.

Another thing to take into account is the geography of the region, with many island and more importantly the fact that these islands were controled by various states meant that it was easy to attack a ship and sell the goods to another nation. Plus the distance between the Caribbean and the states that were supposed to control them and insure law inforcment (England, France, Spain), meant that the central power had more difficulties taking action against pirates.

So overall, the Caribbean was the perfect hunting ground for pirates. But that doesn't explain everything. And that's where Marcus Rediker's book is interesting, because in digs into the social causes of piracy.

Pirates were usualy sailors, that worked on navy ships or marchant ship. But these people were, for the most part, poor people from the coast of England, France or Spain, that didn't chose to be sailors : some were recruited by force (especialy in England, with the impressment system), other were forced to take the job on merchant ship because they couldn't do anything else... Overall, the sailors were not happy to be on navy and marchant ship. Rediker qualify them as an "Atlantic proletariat", meaning poor people deprived of any control over their livelyhood. The suffering of the sailors were accentuated by the low wages and the strong and violent discipline aboard navy and marchant ships.

Rediker explain that, for this people, piracy was a both an act of rebelion against a power they considered injust (the officer on the ships and the trade company), that made a lot of money without giving the sailors a decent living, and an act of self preservation, because staying in the navy or working for a trade company would probably mean death in poverty. So their reasonning was more or less, "Hell, if I'm going to risk my life on the sea, at least let's do it for MY profit and MY sake, and not their."

That a brief summary on how piracy came to be. But again, it's a summary. Reading Rediker's book, or the IAMA thread will give you more details.

 

Why did it died out ? Well, the main reason is that the various states that ruled over the Caribbean decided to stop them. Rediker explained that pirates were a big problem for European state, for two main reason. First, they were a nuisance to trade, and their action wrecked the colonies economy. Second, the political nature of the pirate's undeavour, the rebelion against authority and the fact that they did not recognized any king or state, was a symbolic threat to the European monarchies, at a time they were consolidating their central power, and for France and Spain, creating an absolute monarchy. Having bands of sailors boldly defying any form of authority, proclaiming that the King could go to hell, and that they would die rather than submitting was a big dent in the political construction of these states.

To fight piracy, the states used what Rediker calls "a state terrorism". Meaning any pirate that was capture would by publicly trialed and executed, in the most public way possible, to deter any would-be pirate. Plus they unleashed a full offensive on pirate, sending ships as soon as they could affoard it. Appointing members of the navy as governor, they insure that every representants of the states in the caribbean would fight against pirate. The attack on the Republic of Pirates of Nassau in 1718 is one of the most important blows against the pirates, depriving them of a base of operation in the Caribbean. The military operations managed to fight the pirates.

And it's not suprising : Pirates were not a military force. They were not really organised, and althoug some pirates, like Blackbeard, did control several ship, they were not really a fleat as would be a proper navy. They thrived in the Caribbean because no states had the power to control the area, because it was still being colonised, and not well known. After almost a century, european states asserted their control over the caribbean islands, and therefor needed to eliminate the pirates.

What is interesting - and, in my opinion, sad - is that piracy did not end because the living and working condition on ships improved, not because the need for piracy disappeared, but because the repression was strong enough. And the discipline on ships continued to be harsh and the wages continued to be slim, and sailors were still an Altantic proletariat. Piracy still existed, but after the end of the golden age, it was marginal.

 

Hope I awnser to your question. Again, this is a summary, and in no way an exhaustive awnser about the rise and fall of piracy. Reading Marcus Rediker's work should clarify what I couldn't explain here. I would recommand another one of his book, Outlaws of the Atlantic: Sailors, Pirates, and Motley Crews in the Age of Sail. Plus his book with Peter Linebaugh, The Many-Headed Hydra: Sailors, Slaves, Commoners, and the Hidden History of the Revolutionary Atlantic.