The short answer is that, as part of their accession to the newly formed United Nations, the USSR successfully negotiated concessions from the West wherein certain federal units of the USSR would be able to join the United Nations as full member states. In particular, they requested the admission of Belarus and Ukraine, as well as veto powers for each the Big Five permanent members of the Security Council, including themselves.
To understand why, consider the composition of the early UN: of the 51^1 founding states, those aligned with the West formed a clear majority; outside the USSR itself, only Yugoslavia, Poland, and India (soon enough) could be said to be on the 'side' of the Soviets. While, as we know today, many nations (including China, a permanent member of the Security Council) would soon come to side with the Soviets, this was not obvious at the time; from the perspective of the Soviets in 1945, the concessions demanded were perhaps the only shield they had against the West imposing their own views on the USSR through the auspices of the UN.
Moreover, the concession was not, at least from the Soviet point of view, entirely arbitrary or unjustified. While Stalin's doctrine of Socialism in One Country dampened this somewhat, the early Soviet ideal of a "union of nations" still did persist to some extent by 1945. Under this model, the USSR was a free association of sovereign countries who joined together out of ideological brotherhood -- indeed, Lenin's 1917 Declaration of the Rights of the Peoples of Russia^2 even included a right of secession, one which was quickly (and surprisingly, to the Bolsheviks) exercised by many outlying regions. Beyond that, the Soviet Union did put some effort into 'legitimizing' the independent status of the republics; most notably, the constitutions of both Ukraine^3 and Belarus^4 guaranteed the republics the right to handle their own foreign relations and maintain standing armies, the two features then considered necessary for statehood.
Nor was this circumstance entirely unique. Note that both the Syrian Arab Republic (essentially a French colony, albeit originally under the auspices of the League of Nations) and the Philippine Commonwealth (a territory of the United States) were also part of the first fifty-one states to join the UN^2, though both countries achieved full independence but a year afterwards in 1946. More broadly, consider the Commonwealth of Nations: nations like Canada and Australia, whose governments were hypothetically regulated by a British monarch, were still included in the UN! The idea, of course, is that legal trappings aside they were effectively independent -- which is exactly what the USSR argued, in effect, was the case for their Soviet Socialist Republics.
The exact course of the ensuing negotiations negotiations is somewhat involved. In short, the USSR's initial demand for representation of each of their sixteen states was met by incredulity from the US (who suggested, I assume for the sake of argument, that each of the composite states of the US be admitted in the same way). After protestations from the USSR over the status of the Phillipines and India^5 in the end, the USSR settled for three seats and a veto, a situation which persisted until the breakup of the union.
Nevertheless, the image of 'effective' independence for Ukraine and Belarus was not sustained for long; I won't go into detail here, but in the end the USSR did little to maintain effective independence for its composite republics, and despite its hypothetical structure was in reality a very centralized state throughout much of the remainder of its existence. Nevertheless, with the USSR on the Security Council, there was little chance of removing the two members from the organization -- nor was, to my knowledge, any attempt ever made to do so.
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