Victories in what battles prompted foreign nations to support the Continental army?
The victory at Saratoga New York was the battle that persuaded the French to support the Continental Army.
The French and the English has been at war off and on for decades. The French and Indian wars were just part of a world wide conflict between the French and the English. The French lost.
The French had no desire to engage in another round of conflict that they would lose. However when a total English army was forced to surrender at the battle of Saratoga the French felt that the Colonies had a good chance of winning their war for independence.
Shortly after the British defeat at Saratoga the French agreed to help the Colonies. The French support would prove critical in the defeat of the British at Yorktown. Without the support of the French the British would not have been forced to surrender an another entire army. This lose would cause the British to grant independence to the colonies.