Why did the British issue the Proclamation of 1763?

When the French and Indian War came to an end, the British Empire inherited a great swath of land in the Ohio River Valley. While control over this large territory was one of the leading causes of the war, finally acquiring it did not come without its problems. There were still scattered pockets of French settlers in the region. There were also larger numbers of Native Americans there who had been allies of the French. Even though the European powers had ceased their armed struggle, several Native American groups continued the fight against the British in what was known as Pontiac's Rebellion.
The British were fearful that if large numbers of settlers crossed into the Ohio River Valley, this would prolong and expand the fighting with the Native Americans and possibly rekindle the fight with the French settlers there. The war had been costly, both financially and politically, for the English. Parliament and the king wanted their fragile peace to take root. They considered the movement of English settlers into this newly acquired land to be a possible destabilizing element that could erase the gains they had secured in the Treaty of Paris. That is why King George issued the Royal Proclamation of 1763, which was intended to stop the expansion of American settlers from moving west of the Appalachian Mountains.
Of course, history has shown that the wishes and orders of the British powers were often ignored by their colonists. Even before the Proclamation of 1763 was made, many settlers had already laid claim to lands west of the Appalachians. Many felt that these lands had been hard-won during the war and did not feel that it was right that they be denied to them. Indeed, it was impossible for the thinly-stretched British authorities to enforce the Proclamation, and it was ultimately worth little more than the parchment it was written on.
https://history.state.gov/milestones/1750-1775/proclamation-line-1763

Following the Seven Years War, known in North America as the French and Indian War, the British acquired a large amount of territory from France. Much of the newly gained land happened to be west of the Appalachian Mountains and included farmland on which American colonists hoped to settle. There was, however, a problem with this situation. The problem was the fact that the land was inhabited by Native Americans, many of whom had enjoyed relatively positive relations with the French and had assisted the French in their fight against the British.
In 1763, the British would issue the Proclamation of 1763, which forbade their colonists in the Americas from moving west of the Appalachian Mountains. The British did this because they feared that settlers moving west of the Appalachian Mountains would be encroaching upon lands of the Native Americans, which would further upset them. The British were afraid that the Native Americans would launch attacks against the settlers and thus force them into another costly conflict.
Although the British aimed to prevent conflict through the Proclamation of 1763, it upset many American colonists who had assisted the British in the French and Indian War in order to hopefully gain access to the land west of the Appalachian Mountains. Some colonists disobeyed the order and moved west of the Appalachian Mountains anyway. There were also a number of colonists who were found living west of the Appalachian Mountains and evicted. The Proclamation of 1763 ultimately hurt relations between the British and their American colonists and can be cited as a contributing factor to the growing desire for American independence.

The Proclamation of 1763 was issued at the end of the French and Indian War.  There is some debate as to exactly why King George III and Britain may have issued the proclamation, but the main reason for the proclamation was that it was supposed to protect both the colonists and the Native Americans.  The proclamation itself forbade citizens or colonial governments from buying land or making agreements with the Native Americans.  This effectively shut down westward expansion of the colonies.  By shutting down the westward movement of colonists, Britain hoped to appease the Native Americans who felt threatened by rapid westward colonization.  The thinking was that the proclamation would also protect the colonists from angry Native Americans and their retaliatory raids.  The thinking worked the other way too.  If colonists were not expanding west, the colonists weren't thinking of attacking the Native Americans.  Many settlers defied the proclamation in order to secure more and better farmland, and the proclamation came to an end with American independence.