One possible reason the border created by the Proclamation of 1763 angered American colonists was due to its restriction on westward expansion. They felt its kept them under the strict control of England and that the British only wanted them east of the mountains so they could keep an eye on them.
The Proclamation Line of 1763 was established by the British government after the French and Indian War to prevent further conflicts between colonists and Native American tribes. This line that ran along the Appalachian Mountains prohibited colonial settlement west of it.
This restriction frustrated many American colonists who had fought in the war and expected to be rewarded with access to new lands. They felt that their sacrifices in the war entitled them to the opportunity for westward expansion and landownership and the Proclamation Line seemed to infringe upon their perceived rights and economic prospects.
Note: The map is attached, for better understanding, below.