The Revolutionary War quickly became internationalized because Great Britain was a major world power with rivals who wanted to weaken it. These rivals had a strong incentive to become involved in the war. Their help (particularly that of France) was a major factor in the colonists’ victory.
Any county that is a major power will have other countries trying to tear it down. Other countries will want to weaken that country so that they can benefit from its decline. Therefore, if the major power becomes involved in a war, the other countries will help its opponent if at all possible. We have seen this at many points in our history. The Soviet Union and China helped North Vietnam as a way to weaken the US in the Vietnam War. The US helped the mujahedeen in Afghanistan after the Soviets invaded in 1979 because that was a way to weaken the USSR. In just the same way, France aided the American colonists as a way to weaken its archrival, Britain.
The help that the colonists got from France was probably indispensable in winning the war. The French gave badly needed money and supplies. They also sent over military forces, which were not terribly important overall, but which did help the Americans tremendously in the decisive battle at Yorktown. If the French had not been interested in hurting Great Britain, they would have been less likely to help the colonies and it would have been harder for the colonies to win the war.
The previous post is quite accurate concerning the French participation in the American Revolutionary War. France and England had long been rivals (they would soon go at it again during the Napoleonic wars), and France was glad to side with the Americans because they knew it would weaken England if colonial independence could be gained. By keeping England from controlling the vast Americas, it would weaken them financially, militarily and politically, especially in Europe. The new United States would also give France an important new ally. Once France recognized the colonies' new government, their military and financial support helped to overthrow the British; their fleet of ships that blockaded the Virginia coast greatly aided in the final surrender at Yorktown.
The reason that this conflict came to involve other countries was because England was a superpower and it had rivalries with other countries that wanted to weaken it.
In particular, England and France were rivals. They had fought many wars, including the Seven Years War in 1754-1763. So France was always trying to weaken England. If it could do that by helping the colonies, it would.
French help (and some from the Spanish) really allowed the colonies to win the war. It kept the Americans funded and armed and it was the French fleet that helped trap Cornwallis at Yorktown.