The British and the American colonies won the contest for North America by winning the French and Indian War. Two major reasons are typically given to explain why the British were able to win this war.
First, some scholars argue that the French used up too much of their military strength fighting in Europe. The French and Indian War was only one part of a larger global struggle commonly known as the Seven Years' War. France is said to have concentrated on fighting in Europe, thus making it impossible to send enough troops to fight effectively in North America.
Second, some scholars point to cooperation between the British government and the colonies. The British colonies had many more people than the French colonies did. So long as the British colonists were willing to fight, they could be a formidable force. Late in the war, the British government finally found a way to gain the whole-hearted support of the colonies (by giving them more control of the conduct of the war). This gave the British a stronger force to fight in America.
These are two reasons typically given for why the British won this war.