For some background, this answer I wrote on Soviet democracy and how it functioned might be of interest.

It's probably better to think of these particular candidates as "non-party" candidates rather than "independent" candidates. They were not running with their own independent power bases, or with their own political programs - they were nominated as candidates by the Communist Party and elected in single-candidate elections, just like Party members. The Communist Party very much had a constitutionally-guaranteed monopoly on power until 1990.

These nonparty candidates were usually chosen because they were representative of theoretically independent professional or industrial organizations, such as trade unions or collective farm associations. Such organizations were legally separate from the government and the party proper, but as official, legal entities very much took their orders from the top of the party leadership.

Just to give an example via collective farms: collective farms were on paper non-state, non-party enterprises - basically farmer cooperatives, ie farmers were pooling their productive resources into a single enterprise, and earned a portion of the profits from operating that enterprise. Of course, joining a collective, which had been voluntary in the 1920s, very quickly became compulsory in practice if not in name because of policy shifts from the very top - and those farms also had to sell their collective produce to the state at set annual amounts and prices, so it never really operated as any sort of independent business (the average annual proceeds per farmer in a good year during the first years was usually the equivalent to a loaf of bread).

As a result, collective farms were nominally independent organizations with their own nominally independent associations, but effectively directly controlled by the Party. The Second Collective Farm Charter (which was a template document for how all collective farms should be organized and operated) was adopted by the Second All-Union Congress of Kolkhoz Shock Workers, held in Moscow in 1935. About a quarter of the 1,433 delegates were collective farm chairmen, another quarter farm brigade leaders, and the rest rank-and-file collective farm workers (mostly selected for their production achievements) - basically not Party members. However, they were helpfully provided with a draft charter produced by the Communist Party's Central Committee Agricultural Department, and edits of the draft were handled by a Congress Editorial Commission which happened to be chaired by the Commissar of Agriculture and a humble party member called Joseph Stalin, or something.

Which is all to say, these sorts of nonparty organizations, and nonparty legislative figures, were all stage-managed and existed at the pleasure-to-micromanagement of the Communist Party's senior leadership. Nonparty legislators would be elected in single candidate races by almost the entirety of eligible voters in the same manner other Soviet legislators were elected.

This didn't really change until the 1989 Soviet elections, and the republic-level elections the following years. These were multicandidate elections, but not multiparty elections. Multiple Communist Party and/or Nonparty candidates could run for the same office, and quite a few of the candidates preferred by the Soviet leadership actually lost at the ballot box. Increasingly, politicians and legislators left the Communist Party, especially after it gave up its constitutional monopoly on power, and at this point we can really say there were "Independent" politicians like Boris Yeltsin, who won competitive elections and had their own nonparty power bases.