All of this occurred in a fairly alien world (edit: I should say, temporally remote, far from our UN Charter contemporary time) where territories could be swapped and sold, and claims were strengthened by who "first discovered" a land.
To start, the boundaries of the Louisiana Purchase were not very well defined. Spain and the US were still negotiating over the border in this period.
In 1818, Andrew Jackson outrageously invades Florida (then controlled by the Spanish), on the basis of Florida harboring belligerent native peoples and British (this is after the 1812 war has ended). This was considered an outrage, and most everyone in the US government (and the international community) was furious with Jackson - but John Quincy Adams defended Jackson, as he saw an opportunity. Spain was now afraid they would lose Florida for nothing. Adams uses this as an opportunity in the LA Purchase: the US will buy Florida (so Spain doesn`t lose it for nothing), if Spain will capitulate on their claims over the LA Purchase boundary negotiations (this is the Adams-Onís Treaty of 1819), which includes Spain recognizing the Oregon region as within the US claim. Thus, the US inherits the legitimacy of Spain`s "first discovery", thus strengthening the US claim for Oregon against the British, although its exact boundary with Canada isn't quite obvious.
In the meantime, the British are undergoing the Industrial Revolution, which is largely fueled by cotton textile production. This requires cotton, with US slave plantations are producing more and more of. Thus, while the British had recently fought with the US (don't forget it was a breakaway state on top of this), it was in their interest to normalize relations and trade. The British also had interests on free trade in Latin America and the Caribbean (despite the myth, the "Monroe Doctrine" was, for nearly a century, impotent - Britain dominated Latin America). It was in Britain's interests then to come to terms with America.
So Britain conceded the border along the 49th parallel in negotiations with, you guessed it, John Quincy Adams (in 1818), as well as recognizing the Louisiana Purchase, naval disarmament on the Great Lakes, limiting of British trade with natives on the Mississippi (British-native relations were a sticking point amidst war tensions, as Jackson demonstrates), and some trading rights concessions. Free trade in the West Indies was a sticking point for the US, but economic warfare convinced the US to cave to the British by the 1830s. Edit: however, the 49th parallel agreement got controversial westward through the Rocky mountains. The Oregon country itself was ill-defined, so the compromise was to leave the region open to both British and American settlers (the British insistent on the Columbia river being the border). This was the context for strengthening the American claim in the Adams-Onis treaty.
Edit:
The Hudson's Bay Company (one of those classic mercantile British companies) had by the 1840s established themselves north of the Columbia River (their preferred border), when American fur traders (largely of the American Fur Company) started showing up via a wagon-able route ("the Oregon Trail"), settling south of the Columbia river in the Willamette valley (which today containing Eugene at the south end, northward Salem, to Portland, which touches the Columbia river).
In the wider American picture, Texas was trying to join the union, but as a to-be slave state, there was obvious political ramifications around this (broadly speaking - there had been effort since the Missouri Compromise to enter slave states into the union with free ones), and Jackson and Van Buren stalled Texas' request for annexation in the 1830s for electoral/political purposes. Yet Texas was being flooded with Americans, and with them their slaves, and expansionist tendencies grew with this. The issue (in part due to British efforts to push for emancipation in Texas to subvert attempts by the US Southern bloc to annex it) became a political football during President Tyler's presidency (between rival Democrats Calhoun and Van Buren), a tenure which oversaw Tyler expelled from the Whigs (and themselves losing the midterms terribly, largely over domestic issues). In this splintered context a 'dark horse' candidate Polk - a Jacksonian and expansionist slaveholder - tried to harness jingoism to suppress this sectionalism.
He barely won election, and afterwards, Tyler was able to barely pass annexation, with the victorious Democratic support (US President's have a brief window of time when still in power ("lame duck" period), before the transition of power). This obviously upset the "Missouri Compromise" of balance, and Polk aimed to resolve the tension with an expansionist program w respect to Oregon and California. Polk took a hard line with the British, and his bellicose rhetoric was pushing the British to the 49th parallel (although raising sectional issues of the size of Texas v Oregon, if the full Oregon territory was not brought into the union). To save face, the British Hudson's Bay Company was permitted free navigation on the Columbia river - the treaty was signed in June 15, 1846.
As you might also see coming, as a bonus, this jingoism with respect to both Oregon and California lead to... the Mexican-American War. But that's a different story.
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Hence, the international legalistic framework is in place, supported by economic interests, for the US to claim the territory south of the 49th parallel. (See edited section). Then the Gold Rush happens in 1849, and transcontinental railroads bloom throughout the 1850s-1880s, there are masses of migrants heading west, within the US, beginning to fill the space within the century. Not a trivial amount of time since the John Quincy Adams negotiations, but those formalized the boundaries which were to be "filled in", so to speak. (See edited section)
Sellers: The Market Revolution - Jacksonian America 1815-1846
Hobsbawm: Age of Capital