One result of the French and Indian War was the ceding by the French to the British of all former French territory west of the Allegheny Mountains, essentially the entire Ohio Valley (and more). The British then altered trade relations with the Indians in these territories, which included limiting the trading of weapons and iron to the Indians. Pontiac's Rebellion, an Indian uprising against the British in which hundreds of Americans were killed and thousands lost their homes, is a result of the new trade policies.
When the British settled Pontiac's Rebellion with what is known as the Proclamation of 1763, Americans were not only prevented by the Proclamation from not moving west of the Allegheny Mountains but also those Americans who were still in the Ohio Valley (with some exceptions) were required to move east. Because Americans fought alongside the British to defeat the French, they believed they had a right to the new territories--and Americans needed to expand from the east--but the Proclamation made expansion illegal.
In addition, in order to pay for the costs of the French and Indian War, in 1764, Britain enacted the Sugar Act and, one year later, the Stamp Act, both of which created new economic problems for American colonials and against which they protested vigorously. The Stamp Act is widely viewed as a major American grievance that help lead us toward rebellion.
American colonials viewed Pontiac's Rebellion as the direct result of mis-guided British policy toward Indians, and the Proclamation of 1763, which tried to stop the Indian rebellion, was largely viewed by Americans as a British betrayal of not only their economic interests but also their hard-won rights to new territory in which to expand. Coupled with newly-enacted burdensome taxes, these unexpected results of the war created a breach in relations between Americans and British that helped push Americans to physical conflict in 1775.
One of the foundational reasons as to why the American Revolution was fought resided the outcome of the French and Indian War. Economically, the Colonists put much of their wealth into raising armies for the British cause. They recognized the mutual benefit in driving the French and Indians out of Colonial strongholds. The Colonists also understood that in being able to participate in a war with the British as colleagues, a new level in their own standing as a nation had been reached. The mightiest army in the world required the help of the Colonists to defeat a foe. This has cast a large impression upon the Colonists. In some respects, it meant that the Colonists "had arrived" and were no longer a merely silent offshoot of the British crown.
Economically, the Colonists had made financial commitments during the war in support of the British. This helped to alter the relationship with the British in that the Colonists no longer idly accepted the taxes and acts which would take away Colonial income, funds that had been given to the British cause. In addition to this, the political landscape had changed with Colonial participation in the war. The Colonists were confused as to why they could not be acknowledged as political equals to the British. When acts such as the Proclamation of 1763 limited Colonial mobility, resentment emerged. The same political will that enabled British to keep the land was the same frustration that was expressed when the Colonists could not partake in it. Ideologically, the Colonists never understood how it was equitable or proportional that the costs that the British ensued in the war was passed on to the Colonists' backs, when they, themselves, had made sizable financial commitments in defense of the British cause. It was not clear to the Colonists why they were having to suffer even more for supporting the British in both economic and militaristic senses. These becomes some of the basic reasons why rebellion against the British are sown in the aftermath of the French and Indian War.