I can't answer about homosexuality in particular, but one thing that it's important to realize is that the Soviet Union, despite being led by a highly educated, "progressive" vanguard to begin with, was still largely conservative (before anyone says something, this is not a comment on current politics), with what you would call traditional values, particularly connected to orthodox Christianity. It didn't change overnight just because the on the face of it socially progressive and irreligious communists came to power.

Which is to say, some things did change essentially overnight. After coming to power, the aforementioned vanguard implemented many policies that were for its day incredibly progressive, including but not limited to things like abortion, child care and rights for women.

This only lasted a few years though, because it quickly became clear that the rest of the party, let alone the rest of the population, wasn't really interested in some of these new changes. When Stalin came to power he quickly reversed (many of)* the policies that were still in effect. In some ways Stalin was a reactionary communist, and he wasn't interested in instituting policies that were largely unpopular and didn't provide any tangible benefit to his control over the party or the country.

You question touches on two of my biggest pet peeves when it comes to Soviet history: people assume that Bolshevism sprung ex nihilo, instead of inheriting huge social problems from Tsardom, and they treat Soviet history as monolithic, instead of a acknowledging it consisted of very distinct eras. The criminalization of homosexuality in the USSR has to be understood as a product of inherited prejudices, pseudoscience, and the culture of Stalinism.

In Tsarist Russia, the Orthodox Church had condemned homosexual acts for centuries, but only as late as 1832 did Emperor Nicholas I pass a law criminalizing sodomy outside the army. This law would prefigure statutes passed by the Soviets by ignoring lesbianism and conflating consensual sex with rape or pederasty. However, the Bolsheviks were also impacted by other currents of pre-revolutionary thought. German (Freudian) psychology was hugely influential in leading Russian intellectuals to conceive of homosexuality not as a sin but as a sickness. Openly gay writer of the Silver Age such as Mikhail Kuzmin and Marina Tsvetaeva contributed to perception of homosexuality as a perversion of the decadent and foreign-influenced intelligentsia.

With the triumph of the Revolution, many traditional attitudes were challenged publicly or surreptitiously. The new criminal code of 1922 decriminalized all forms of consensual sexual activity. Prominent Bolsheviks such as Alexandra Kollontai participated in the World League for Sexual Reform, a Germany-based organization that sought to decriminalize homosexuality and demolish misogyny on a scientific basis. Supporters of the social hygiene movement saw homosexuals as victims of a disorder, who should be treated with compassion and left alone. Others were even less critical. A college student named Grigorii Batkis published a pamphlet entitled "The Sexual Revolution in Russia," in which he enthused about the new sexual freedom in Russia, which at least on paper surpassed that of the rest of the world.

Still, wide-spread distrust and dislike of homosexuals remained. The revolutionary code of ethics discouraged individualistic pleasure-seeking, which conflicted with the rigid selflessness necessary to survive as a professional revolutionary. This attitude was epitomized by Nechayev's Revolutionary Catechism: "All the gentle and enervating sentiments of kinship, love, friendship, gratitude, and even honor, must be suppressed in him and give place to the cold and single-minded passion for revolution. For him, there exists only one pleasure, on consolation, one reward, one satisfaction – the success of the revolution." Citizens could be inspired by this strict puritanism as well. In the revolutionary optimism of these days, many scientists believed perversions would disappear along with other social ills. The psychiatrist V.M. Bekhterev argued that homosexuality could be cured by hypnosis and prevented by a proper upbringing. The permissiveness of Soviet law reflected a certain degree of indifference to individual shortcomings, that would soon be a thing of the past. As another psychiatrist, Lev Rosenshtein, put it, "We are not so interested in individual misfits as such. We are interested in making a life that will not produce misfits."

These attitudes combined with the rising paranoia of the thirties to produce a crack down, characterized as a "Sexual Thermidor" or "the great retreat." In 1933 male anal sex was re-criminalized, although not lesbian acts, showing the persistence of Orthodox attitudes towards sex. Procreation was encouraged to deal with a falling birthrate and the increasingly serious threat of war from the West. This militarization of society saw a popular backlash against many of the libertine attitudes of the earlier period. Homosexuality was seen as a symptom of bourgeois individualism, putting a Stalinist twist on Lenin's warning not to engage in promiscuity: "Dissoluteness in sexual life is bourgeois, is a phenomenon of decay." Maxim Gorky equated homosexuality with fascism in the misnamed essay "Proletarian Humanism." As the Motherland demanded everything for her defense, homosexuality was seen as a dereliction of duty. Consensual sex was again conflated in the public mind with antisocial acts such as prison rape and pedophilia.

That de-Stalinization never tackled this issue is a reflection of the weakness of the gay community in Russia, wide-spread bigotry, the persistence of pseudoscience in Soviet psychology, and international acquiescence.

A lot of this is a shameless paraphrase of "The Russian revolution and the Decriminalisation of Homosexuality" by Daniel Healey.

I know some information about this topic; I might be able to provide a basic answer to tide you over until a true Russian history expert can answer more completely.

Procreation was indeed one of the major factors behind it, although it would be difficult to prove that as the definitive reason for its illegality, and there were certainly other reasons behind the move. Initially homosexuality was legal in the Soviet Union, but as time ground on and Stalin assumed power the Union gradually became a more and more conservative institution socially. Stalin maintained many of the progressive changes of the earlier Soviet Union, but also rolled back many changes he believed were damaging to the fabric of society. He was particularly hard on women's rights, for example, and attempted to push a more family-oriented role with women, particularly in regards to children. This, along with a slightly more lax view on the people's ties to the Orthodox church (especially during the war years), is oft cited as the beginning of the Soviet Union's backstepping into a socially conservative but politically leftist position.

Unless I misremember, homosexuality was made illegal once again in the mid-30s, roundabouts the same time Stalin was rolling back some of the earlier rights women had achieved--including the right of abortion--and pushing for a more family-oriented policy. The move was no doubt in part to try homogenizing this new (or should I say old?) family unit Stalin envisioned for the Soviet Union's basic social building block, and undoubtedly was also done in the hopes that it would promote greater population growth, potentially with the goal of helping to boom industry. The population growth concept is particularly compelling, since abortions were made illegal about the same time as homosexuality was. Whether or not the move could be attributed to a general conservative trend in the Soviet Union is something for greater research into the period than I have attempted--I personally would claim much of the conservatism of the time came from Stalin (who should not be mistaken for conservative on all fronts, but certainly was on some), since he was effectively the one making all of the decisions, but I'm not the final authority on population trends by any means, and it is certainly true that the population of the Soviet Union was far more conservative (at the time) than its leadership.

As for why this change wasn't rolled back, that's more difficult to answer. It could very well be that the Soviet Union always viewed itself as underpopulated compared to the dense, high-population enemies (particularly the USA) that it considered itself to be facing, and thus believed that homosexuality being legal would damage its ability to grow its population proportionally to its foes. It is also possible--and this is what I view to be probable--that the socially conservative policies that Stalin began grew and multiplied during the time of Khrushchev and, in particular, Brezhnev, and thus there was no motivation for these essentially socially conservative peoples to re-institute a policy that they personally disagreed with. This analysis, however, perhaps oversteps my understanding of the period.

Source: Ronald Grigor Suny, The Soviet Experiment

Just to clarify, homosexuality was decriminalized by the USSR (specifically just Russia, not he satellite states) in 1922, and then Stalin re-criminalized "muzhelozhstvo" in 1934. So technically the Bolsheviks were initially extremely progressive in LGBT areas. Stalinism reversed those progressive policies.

Hazard, John N. Unity and Diversity in Socialist Law.

Another point: there is an opinion that the criminalization started with the head of OGPU (an organization tasked with fighting counterrevolution and spies) filing a report (russian wiki, sorry) about homosexuals forming underground groups susceptible to recruitment by foreign spies.

Castro is on record stating that homosexuality is a "bourgeoisie disease". Engels was the source of this notion. Lenin repealed homosexual laws, but Stalin reinstated them.

Linking homophobia exclusively to religion is a fallacy that many people make.

Throughout history homosexuality has been often viewed as taboo/wrong. Regardless of religion. This is often why it is found as such in religion, not the other way around.