Why did King George III issue the Proclamation of 1763?

Two main reasons: keep the American colonists closer to the coast and therefore easier to control, and to cut down on contact with French and Native American populations on the other side.
1763 saw the end of a worldwide war between France and Great Britain. On the American continent, that war was waged primarily in the mountains between British colonists and the British army on one side, and the French army and colonists, allied with Native American peoples on the other.
After the war concluded, the British crown wanted to prevent the repeat of such an expensive conflict, and decided that the American colonies were growing too quickly. They had stretched well into the interior of the continent and were far away from the centers of British power near the coast. The Proclamation of 1763 attempted to limit that expansion but was immediately unpopular and proved to be nearly impossible to enforce.
It was an early source of dissatisfaction with British rule that would simmer and grow until the outbreak of the Revolution