The answer above covers it pretty well. Here are a couple of other points to consider. The French and Indian War lasted for nine years, and was expensive to the tune of $15,000,000. The King and the British people wanted it done and over with. So by defeating the French and negotiating the Treaty of Paris, that took care of one enemy, but the tribes on the frontier who had fought alongside the French had also been armed by them, and remained a threat.
So the King wanted to buy them off with guarantees of land, west of the Appalachians. The Proclamation was that guarantee. Most of the people living there were poor, former indentured servants and late arrivals to the colonies, who owned no slaves and worked small family farms, often living off the land as frontiersmen. These were Scots-Irish, and they had no love for the King.
So they ignored his order and settled west of the line anyway, causing more tension with the native tribes there.
The proclamation of 1763 was meant to reduce the tensions that were happening between the colonists and the Native Americans. Basically, it kept settlers from moving beyond the Appalachian mountains and left that land for the natives. You can see a picture of that line drawn here. It also had the effect of keeping the colonists concentrated along the Atlantic coast where they would be easier to regulate and control.
The settlers were slowly moving west and running up against the natives already living there. Tensions between Britain and these tribes were very high and the government wanted to keep the "warring" parties separated.
The colonists, to put it mildly, were cheesed-off. The "French and Indian War" had just been fought and the big part of the prize was gaining all that territory from the Frenchies. To have that prize denied them was infuriating. Worse, a series of forts was to be built to keep the settlers out...at the expense of the colonists.
So there you have it. The Proclamation of 1763 had the hope of reducing tensions in the colonies and making them easier to control, but in the end it created as many problems as there were before.