We might wonder why the British were willing to allow the independence of the United States in the 1783 Treaty of Paris, as well as US territorial extension to the Mississippi, navigation rights on the Mississippi, fishing rights in Newfoundland, and especially the withdrawal of British troops. The answer involves US military victories, British conflicts with other nations, and instability in British politics.
First, the US victory at Yorktown in 1781 led to British openness to US independence. Clearly, the Americans were not inclined to give up their fight, and they were beginning to turn the tide militarily. When the British under General Cornwallis surrendered to George Washington after heavy losses, the British knew they couldn't keep up the fight much longer.
Further, Britain was entangled in conflicts with France, Spain, and the Netherlands, and the British were spread thin. These nations were at least unofficial allies with the US, and the British hoped that peace with the US (complete with the recognition of US independence) would open the door to negotiations with the other countries and perhaps peace settlements with them as well.
Finally, the British government was rather unstable during these war years with, Parliament and the king at odds and a succession of government leaders unable to settle difficulties. Finally, Lord Shelburne took the reins as Prime Minister. He was open to peace but hoped to avoid US independence. War, however, was proving to be extremely expensive, and Shelburne had to consider the conflict on several fronts as mentioned above.
By November 30, 1782, US negotiators Benjamin Franklin, John Jay, Henry Laurens, and John Adams made a preliminary agree with Britain that depended upon peace between Britain and France. When this was finally concluded, the peace treaty recognizing US independence could finally be formalized on September 3, 1783.