Louis XVI was probably a bit conflicted about the prospect of helping a country overthrow their sovereign Lord. He didn't really want his own people to get those kinds of ideas, though in the end he couldn't prevent it. And that was partially because wars, including the aid they contributed to the American Revolution, ended up ravaging France's treasury and making him look like a foolish puppet for scheming advisers that led him into losing ventures that weakened France economically and as a world power.
MOTIVATIONS FOR FRANCE TO BECOME INVOLVED IN THE FRENCH REVOLUTION
1. Assert themselves militarily
The main motivation for France's getting involved in a potentially dangerous and losing war venture was ironically that they had just come from several and wanted to reassert their strength, both in Europe and in the New World. They had just had disastrous results in several painful drawn-out wars, the French-Indian War, for one, to compete with Britain and their future allies, the colonists, for land in North America.
They lost this war, of course, which is evident today merely by the fact that basically North America Speaks English and South America Spanish. France ended up rather elbowed out of the perceived booty in the colonial game.
On a slightly larger scale, so much so that it was often referred to as a 'World War,' involving itself in another try to curb Britain's success at colonizing North America (that few in Europe were very thrilled about), they entered The Seven Years' war which seemed like it would be a cinch because Britain had few and rather uninvolved allies. But they were sent home with their tail between their legs yet again.
So by this point France's foreign and military policy was looking rather weak. They had had rivalry with Britain dating back to the thirteenth century, it is true, (though it is debatable that they would have entered the war just to have a chance to fight them, they had that chance constantly and mostly declined). They wanted one last chance to prove that they were not militarily laughable and protect what holdings they did have in North America so they wouldn't have to keep selling them off at cents on the dollar just to finance their treadmill's progress militarily.
2. Search for new friends and allies
At this point the colonists were looking like better and better allies than any they had garnered in Europe with their continual against-the-odds military victories. Most of the rest of Europe had few stakes in the actual game, and if they passed on the Swiss or other countries they would not have soldiers needing their payments, and in fact they as aid were hoping to be payed off with land when they helped them to victory against their old enemies the Brits.
Even if they didn't end up with North America it might be better to have it as a friend than an enemy, and there wasn't any centuries' old rivalry that prohibited the alliance, either borne of prejudice or religious differences. The colonies were home to a hodge-podge of disaffected Puritans and Catholics alike and it wouldn't look like they were either warring against Rome or starting a religious crusade.
And in fact the French did learn quite a bit from their experience with the type of modern warfare that the colonists were forced to practice. Americans couldn't afford shiny new uniforms or other luxuries that traditionally won wars, so much like Britain against the Spanish Armada they employed more guerrilla tactics than the gentlemanly formal lines of battle that Europeans were used to. By the time the French were done they practically took on the whole of Europe and now was dominant, once during the Revolution itself and then during Napoleon's reign, and were the better militarily for their experience in the Americas.
3. Financial advantage at home and the New World
An alliance with the colonists also may have had some side benefits closer to home like taking the Port of Portugal (part of a marriage dowry dating back to when Portugal became Britain's ally in the 1600's when Charles II married Catherine of Modena). Their theory was that when the Isles had their hands full they might be able to enrich themselves at home as well as abroad. Their country was struggling economically, as became poignantly clear later in the century, and they contemplated many such back-door strategies to get cash.
4. They were asked
The colonists were seriously strapped for cash, having any existing aid from Britain cut off and very few allies since pretty much everyone they turned to would be a wolf in sheep's clothing. The rest of Europe was loosing interest, they were afraid of Spain, and so France, a natural choice with land treaties forming and a long-standing enmity for Britain well known.
DIRECT IMPACT OF THE FRENCH ON THE WAR
1. Established part of the need
Many overlook that though France lost the fist few skirmishes in North America to Britain, they did seriously help to deplete the British treasury, making the infamous tea tax among others necessary, leading the colonists to revolt on a large scale. So it is very likely that one of France's first roles that led to Britain's loosing the colonies was to help in necessitating the war for independence in the first place.
2. Gun powder
Swords and rabble rousing would be no match for the British guns and cannons - they needed to match their firearms. And because the colonies was not capable of manufacture themselves, they greatly benefited when France sold them large quantities for a good deal.
3. Assisting American piracy
Britain, who used to dominate the Atlantic stealing Spanish gold, found that with French help, had the tables turned on the mother land by the Colonies who claimed some much needed funds by stealing it directly from their enemies with French backup. America had strengthened its navy by needing to travel across the Atlantic so many times and up and down the long coast, so with talent and assistance they were able to secure much needed funds that without, there would be a lot MORE stories about General Washington leading shoeless troops.
3. Eventually open battle
France was finding its aid to the Americans more of an all-or-nothing affair than they had initially planned. Technically they had other allies in Europe that they had to neglect and turn down when asked for help and it was causing the rest of Europe to loose good feeling for France. The animosity of Spain to the Colonies was mutual, since there was much to be done about where to draw borders so the last thing they wanted was to aid their potentially future competition. (They eventually assisted as it looked like Britain might be brought to its knees and they wanted part of the action, not for any love of the Colonies).
So while the French soldiers were willing (again, not as willing as the Monarchist government, strapped for cash and not entirely on board in theory), it was decided that they could put their toe in the water for actual military combat across the Atlantic. Diplomats like Franklin were dispatched to help rally for the cause he said as part of the Enlightenment philosophy that France had been so key in nurturing. So gradually French troops were sent to the Americas and with enthusiasm. General La Fayette, eventually one of France's biggest military giants, began as a volunteer enlisted man to help the colonies.
The overseers of the navy were right to worry because Britain was waiting for them all over the Atlantic. But battles on land went well and the French people, ready for revenge on England, were enthusiastic, so the government decided to increase their support. While the British fleet was kept busy general Cornwallis was trapped between the Americans and French at the battle of Yorktown, a decisive victory for the colonies.
DID FRANCE ACHIEVE ITS AIMS?
France did regain some of its military pride after the war, perhaps leading to future victories in Europe. But ultimately, neither the Colonies nor Britain wanted France to have any of the land up for grabs, and neither did France's ventures in Europe like the storming of Gibraltar or India come to much. So while to end the war, the secret peace treaty that Britain and the U.S. signed left France out in the cold after all the aid and money they spent. What was worse, Great Britain immediately became the main U.S. trading partner, a reward France was almost sure of.
The disaster to France's economy and the victory overseas perhaps both emboldened enlightenment thinkers there and nearly bankrupted the country, perhaps leading to the Revolution there, as conservatives were frightened and shut the middle class out, leading to hostility. And their friends the Americans didn't help the government officials, once their friends, who found their heads on the guillotine.
So did the French intervention help France like all the calculators thought it would? It depends on who you ask. The royal family found themselves imprisoned not too many years later, the people blaming Marie Antoinette for the deficit because of some inconsequential trinkets compared to the cost of war. And of course during the war they had turned their back on Austria and other allies who could have helped them, so they, like the colonists once were, found themselves very much alone, while America, with their help, was beginning down the path to be the world power that each of the European nations were in their time, but not having to use absolute power to keep it.
AND HOW DID AMERICA FARE?
A third world nation not much more than 200 years ago now America is truly blessed. And that is precisely how the Colonies looked at it. They felt they had been smiled on from above at last after the repression they had endured elsewhere. There were times where all of the founding fathers believed they were sacrificing their lives for a useless cause but believed in it too strongly to give up. Only a man who has the power to be a king can give a throne back to the people, and that is what Washington did.
Washington's people were ready to elevate him to king status but the exercise of the revolutionary war, though perhaps using Machiavellian means at times, was one in freedom and democracy that they all, even those who could have seized power on American soil at this point for themselves instead of dying in near obscurity like most of them, believed in. While the dictators in Europe had to be ripped from their thrones, freedom was then and remains today America's most cherished value thanks to whatever help or luck we could get, and fortunately that will be a hard state to change now.