For starters, it is very unlikely that Louis XIV, who died in 1715 immediately prior to the formation of the Banque Generale Privee and the Compagnie d'Occident, had anything do to with the subsequent crisis, and even less likely that his long-deceased father did.

The Mississippi bubble is best described as a radical attempt to reorganize the French monarchy's finances, but which was far too ambitious, and ended up as little more than a dodgy scam. Its protagonist, the Scottish economist John Law, was a brilliant man with revolutionary ideas that anticipated modern central banking in profound ways. He is also a scoundrel who initiated a disastrous set of reforms for which the French tax base and financial system were utterly inadequate, tried to wallpaper the whole thing over with what amounts to a confidence scheme, and had to flee for his life following its collapse. The story closely resembles the South Sea Bubble in England, though vastly more ambitious in its aims.

The underlying radical idea of Law's system is that the government can issue fiat money on one hand, paper notes not backed by anything in particular, and can balance that issuance by buying and selling the government's debt, which is funded by taxes, which are denominated in that same paper money. This follows on from several centuries of fiscal and monetary innovation in Northen Italy, in Spain, and in the Low Countries, and does not just come out of nowhere. But nobody went as far as John Law.

This astonishing feat was possible because Law was given power (astonishing amounts of it) by the Duke of Orleans, the regent of France, who was looking to put the monarchy's finances on a better footing after the War of Spanish Succession, and to make use of credit as a way to expand the state's spending capacity in the event of a future war. He first gave John Law the right to set up a private bank, the Banque Generale Privee, which could issue its own bank notes. The value of these notes was reinforced by a government decree that they would be accepted at face value in payment for taxes, and then demonetised the old silver mint coins, which effectively made the bank's notes the legal tender of France. Shares in the bank could be purchased by exchanging French government debt. In December 1718, the Banque Generale Privee is effectively nationalised, and becomes the Banque Royale. John Law is now the king's banker, and the notes of his bank are the currency of the land.

So, that's the first part, the establishment of the bank. Why then the "Mississippi" angle? Alongside setting up the Banque Generale Privee Law also started another company, a colonisation scheme for the enormous-but-not-very-inhabited French territory in Louisiana. (This is not just the modern state, but the whole area that would later be purchased by the US.) The King had regained direct ownership of the territory from its previous owner in payment of taxes. This was originally just the usual Early Modern colonisation and trade scheme, along the lines of the various East and West India Companies, or the Virginia Company, or the Hudson's Bay Company. Just like with Law's bank, you could purchase shares of the new company by converting government debt. And so, you ended up with a company, the Compagnie d'Occident, whose balance sheet was basically just a whole lot of government debt, and a territory in North America. John Law himself bought up quite a lot of the shares in the company, as did the King (well, the king's financiers, as Louis XV was about eight years old.)

Law proceeded to use the Compagnie d'Occident to purchase other government tax farms, monopolies, and colonisation schemes. As the company grew bigger, it made successive share issues to raise further funds. He leveraged the company as far as it would go, selling off much of the royal debt to buy further tax revenues. So now Law runs two companies: one bank that effectively prints the sovereign currency, and another that owns large shares of the tax base of France.

In 1720, the whole plan comes together. All tax collection was centralised by Law (and also by law) into one company, with Law now acting as finance minister for the whole country. The government's creditors were given the option to convert their government debt into shares of this company - effectively becoming equity holders in the state's finances. This one giant entity effectively became both the tax agency and central bank of France. For another, Law broke the link between metal reserves and money creation, effectively allowing the bank to print as much money as it wanted, and so long as Law was willing to lend to his other company (i.e. to himself), the company was essentially able to do the same. Metallic money was made illegal, metallic payment clauses in contracts were voided, and the import of gold and silver was forbidden. If you wanted to do business of any kind in France, you had to do it in bills issued by Law and denominated in livres tournois, though there remained a pesky clause that those bills themselves could be redeemed at the Banque Royale for silver, a clause which ultimately leads to Law's undoing.

Law's plan has come to fruition; the metallic standard of France was no more, taxation and royal debt have been centralised, and his magnificent system stood not just at the heart of French finance; it was almost the totality of it. It is important to note: this system is not conceptually flawed. Indeed, it is in some ways far, far ahead of its time, functioning very much like a what exists in the late 20th century in most developed countries. But it is a very radical departure from established ideas of banking.

Law's scheme, however, required that investors wanted to hold shares in his company. If the price of the shares dropped, then investors would stop trading their old royal debt to the company. This left him in a bit of a dilemma; his exponential expansion of the company was almost ludicrously leveraged, and the bank's bills were circulating in greater numbers every day. They tried to keep up the share price by buying up shares themselves. But, of course, since those were newly printed bank bills they were using to buy the shares, the value of the bills started to drop, leading to inflation. Law tried to reverse course, stop issuing bills, and stop lending freely to the King, but this caused the share price to tank, causing pressure from the other side.

The whole mess goes back and forth a few times, and the details are quite complicated, but the situation spirals out of Law's control. The question of whether the notes would be ever repayable in silver, and at what rate, became urgent. Law returned to trying to soak up all the notes issued by selling company shares and issuing government bonds. (The number of schemes and conversion regimes is baffling and ever-changing, reflecting the incredible difficulty Law was facing in stabilising his system.) By the end of 1720, metallic contracts were re-legalised, and the system of using bank notes as legal tender had collapsed. The value of the remaining notes fell to nearly zero.

John Law fled France late in 1720. The government proceeded to clean up the mess by handing over control of royal finance to his rivals. Louisiana never amounted to much of anything as a French colony, and would be sold off by Napoleon to fund his wars. The company was in any case never more than a sideshow in the much larger project of redesigning the entire French monetary and financial system.

Sources: The narrative here is largely from Francois Velde's paper "Government Equity and Money: John Law’s System in 1720 France." For the international dimension, Larry Neal has written many fabulous works on the early bubbles and the functioning of finance in this period more generally.

As for the second part of the question, whether this had anything to do with the French revolution, it is almost impossible to say with any certainty across seventy years what might have happened. The finances of the French monarchy were clearly important as a cause of the revolution. The problem was that the King spent too much (and in large, unpredictable chunks) and taxed too little, and in inefficient ways. If John Law had succeeded, who knows what state they would have been in. But the status quo returned relatively quickly after the bubble, with state finance being neither particularly worse nor better than before. So no, the Mississippi bubble is only a cause of the French revolution in very indirect "what if" sorts of ways, though they are both related to the core problems of government taxation and finance.