When we're specifically talking about Kyrgyzstan and Ukraine (most of it at least, not Galicia and Volhynia), these are territories that were controlled by the Bolsheviks from the end of the Russian Civil War onwards. Eastern European countries like Poland and Bulgaria were not under Soviet control until 1945.

As far as how the Soviets came to control Eastern Europe, I would humbly recommend this longer answer I wrote here. The long and short is that the Soviets were interested in making sure they had friendly regimes in Eastern European countries, but did so via the framework of Inter-Allied Wartime Agreements like Yalta and Potsdam. The only countries that the USSR straight-up annexed after 1945 were the Baltic Countries of Estonia, Latvia and Lithuania: the Soviets claimed these were voluntary requests for union and largely did so because they had been part of the Russian Empire for a long stretch of time, and had particular strategic importance (the Baltic coast). But even here the United States and most other countries did not (and would not) recognize these annexations as legitimate. All the other annexations were border adjustments with countries that continued to exist.

As for the Soviet Union itself: the Bolsheviks ended the Russian Civil War effectively in control of several nominally independent countries: the Russian Soviet Federative Socialist Republic (RSFSR), the Ukrainian Soviet Socialist Republic, the Belorussian Soviet Socialist Republic, and the Transcaucasian Soviet Federative Socialist Republic (TSFSR), which was made up of the Armenian, Georgian and Azerbaijani Soviet Socialist Republics. There was also a Bukharan People's Soviet Republic and Khivan People's Soviet Republic in Central Asia, in former Russian Imperial Protectorates, and a Far Eastern Republic in the Russian Far East (this was effectively a buffer state that existed because of a Japanese military presence - once Japanese troops withdrew in 1922 the state collapsed and was absorbed into the RSFSR).

Anyway, these territories were functionally controlled by the Bolshevik Party (and its Red Army) despite being nominally independent, and in 1922 the question was how to administratively reorganize these territories. Stalin (the Commissar for Nationalities) advocated that they all be absorbed into the RSFSR, but Lenin advocated for a federal model, and this view ultimately won out in the months of talks in 1922. In December a Union Treaty was signed by delegates of the Ukrainian SSR, Belorussian SSR, RSFSR and TSFSR that formally established a Union of Soviet Socialist Republics. Khiva and Bukhara joined the following year, and in 1924 the Central Asian territories of the RSFSR were reorganized, with a Turkmen SSR, and Uzbek SSR created, as well as a Tajik Autonomous SSR in Uzbekistan and a Kazakh Autonomous SSR and Kyrgyz Autonomous SSR in the RSFSR. These latter republics were later elevated to full Soviet Socialist Republic status. The TSFSR was likewise dissolved at this later time and its constituent republics also became full SSRs.