The border is related to early exploration & colonization by Europeans and the 1825 Treaty of St. Petersburg between Russia and Britain.

In the late 1700s, Russian, British, and Spanish explorers began sailing to what is now the coasts of Alaska, British Columbia, Washington, and Oregon. Russian explorers reached as far south as British Columbia where they were met by the Spanish. British Captain Cook later sailed as far north as Cook Inlet at present-day Alaska, but found a Russian presence already established. These early explorers were trading with aboriginals but also hunting for themselves the local fur-bearing sea mammals. There were no proclaimed territories as of yet, and the British later described the situation:

The events of 1787-88 must have been puzzling to the natives of Prince William Sound. Englishmen under the English flag, Englishmen under the Portuguese flag, Spaniards and Russians, were cruizing about, often within a few miles of each other, taking possession, for one nation or the other, of all the land in sight.

In 1799, the Russia-America Company was formed with the consolidation of several fur trade companies and was tasked by the Tsar with colonization and fur trade in Russian America. Russian America was proclaimed, by the Tsar, as all lands of America extending to the 55th parallel. However, over-hunting led the Russians to search for new resources in minerals, timber, and land mammals in the early 1800s. Their presence extended to northern California, and they even attempted to establish a presence on the Hawaiian Islands.

Another ukase was issued by Emperor Alexander in 1821 which claimed for Russia:

the whole of the north-west coast of America, beginning from Behring Straits to the 51st of northern latitude.

The ukase also attempted to exclude foreigners from 100 miles of the coast. By this time, Spanish presence on the coast was non-existent as they were battling revolutions in their American colonies. The United States were present in their place.

In the early 1800s, the United States and Britain were in dispute over the Oregon Territory (U.S.) or Columbia District (U.K.). The area could generally be described as present-day British Columbia and Washington State. The United States signed the Russo-American Treaty of 1824 defining the northernmost U.S. claim as the present-day Alaska-British Columbia southern boundary at 54°40′ north. Britain then signed their own treaty with Russia in 1825, the Treaty of St. Petersburg, which read the following:

the said line shall ascend to the north along the channel called Portland Channel as far as the point of the continent where it strikes the 56th degree of north latitude; from this last-mentioned point, the line of demarcation shall follow the summit of the mountains situated parallel to the coast as far as the point of intersection of the 141st degree of west longitude

After the Oregon Treaty of 1846 established the Canada-U.S. border at the 49th parallel, Alaska was sold to the U.S. for $7.2 million USD in 1867. The Colony of British Columbia then joined the Dominion of Canada in 1871. Canada wished to define the border in their favour - to give them access to the open sea at points north of Prince Rupert. The U.S. and Canada went to arbitration, settled in 1903, to strictly define the panhandle. In the end, one British tribunal member swung the vote in favour of the U.S., defining the border we see today. Canada refused to sign the decision.

Sources:

Lightfoot, K.G. (2003) Hist Arch 37: 14. https://doi.org/10.1007/BF03376620

Great Britain (1893) Behring Sea Arbitration https://books.google.ca/books?id=EusnAAAAYAAJ&source=gbs_navlinks_s

Farr, D.M.L. (2006). Alaska Boundary Dispute http://www.thecanadianencyclopedia.ca/en/article/alaska-boundary-dispute/