King George III issued the Royal Proclamation of 1763 on October 7, 1763, which outlined the division and administration of the newly conquered territory, and to some extent continues to govern relations between the government of modern Canada and the First Nations. Included in its provisions was the reservation of lands west of the Appalachian Mountains to its Indian population

You have to remember that England had finite resources. After winning New France they mostly wanted to get the troops back home and didn't want a bunch of colonials starting another conflict with the native tribes. Although by withdrawing troops the Colonials felt quite comfortable ignoring order from across the ocean and weren't so interested in paying the taxes that were ostensibly to pay for the war the british just fought for them. Some say this is when the revolution really happened as the people began to ignore the kings laws as they pleased.

Relations with natives were very, very strained. Britain had just concluded one of its most expensive wars of the century. They sought peace with the French's former Indian allies. However, it was a very uneasy peace as many Indian tribes (such as the Delaware, Shawnee, and Ottawa) held a huge distrust of the British. British colonists encroaching on Indian lands didn't make it better either, especially since that was one of the big factors of Indian resentment. One could argue that expansion would have benefited the British, but that would mean to completely ignore the Indians as a player, which would have been impossible. the British also needed the Indians for the expertise, trade, and subsequent (and legal) land speculation. For the British, it was a better decision than going back to war all over again with the Indians.

That being said, the Proclamation of 1763 didn't prevent every squatter from crossing the boundary...

For a general overview of this go check out Fred Anderson's Crucible of War, this provides you with necessary context of the period. I believe he also has another book that is much shorter and concise. The French and Indian War was a very important aspect of the politics to come, especially the American War of Independence. Also: Colin Calloway's Scratch of a Pen to further explain the importance of the end of the war. It get a little more in detail about it than Anderson.

I remember reading a line of reasoning that the iron producers of Pennsylvania wanted to expand their operations west, and were resentful that they weren't allowed to (Lancaster County folks were already looking west toward the Pittsburgh region). These were the people who then went about creating implements of war, smuggling in gunpowder, and fashioning a fleet of river gunboats to control the Delaware River.

To these Pennsylvanians, the motive was clearly to keep the colonies subservient to British industry. The Pennsylvanians were only allowed to export pig iron to Britain, where it would then be refashioned into industrial goods and sold back to the colonies. Such unequal economic relationships existed in most other marketplaces, which led to smuggling operations and the widespread usage of flax for clothing.

But. That's just the Pennsylvania perspective. The native american issue seems to have been much more prevalent elsewhere, Virginia in particular.