The French and Indian War started because of the fur trade. During this time the global balance of power had already shifted. Both British and French wanted to extend their colonies into the Appalachian Mountains. During that time, both countries already had fur traders doing business with Native Americans.
But the French believed they had exclusive rights to the land because their explorers were there first. For this reason, the French forced the English out by capturing several of their trading posts and destroying and Indian village that supported the English traders. It was called the French and Indian War because both the British and French armies fought with their respective Native American allies/colonies.