The French and Indian war grew out of territorial disputes between the British and the French in what was then the western edge of the British colonies. The French and British disputed who owned what in the Great Lakes region, in the Ohio area, and along the Mississippi, as well as in Canada.
Fighting broke out during the Seven Years' War. At this time, Britain had twenty times more settlers on the North American continent than did the French, and Britain had thousands of troops stationed in the colonies. The much smaller French population relied heavily on alliances with Native Americans for defense.
After the war broke out, the British were much more willing to devote significant military resources to the Americas than the French were. The French, tied up with fighting the Prussians in Europe, were loathe to divert resources from their European fight to their American territories. Outgunned and needing to focus on their more pressing issues in Europe, the French decided to cut their losses and sue for peace. They ceded territory to the British, which consolidated British power in North America.
Ironically, it was the British willingness to spend a huge amount of money expelling the French that led to the American Revolution and the loss of British control of the colonies south of Canada. After the war, the British wanted the colonists to help pay off the war debt, so they increased taxes and tariffs. This enraged the Americans, who, with the French threat gone, no longer had any need for the British.