To answer your question you have to understand a few things about Marxism.

Marxism is centered around dialectical materialism (diamat). Marx began his academic life as a member of The Young Hegelians, basically intellectuals subscribing to Hegel's views. Hegelian dialectics viewed history as the interplay between ideas; this is a simplification, but basically, opposing ideas clashed and the clash in turn produced the world as it is. Marx came to reject this idealism (the belief that ideas create reality) in favor of materialism. In Marxist diamat, economic circumstances and the socioeconomic conflicts arising within society are responsible for the movement of history. As The Communist Manifesto boldly declares, "the history of all hitherto existing society is the history of class struggles." (FWIW older Marx would often temper and modulate the beliefs of young Marx; for example, Critique of the Gotha Program almost reads like a critique of a program very much like the Communist Manifesto itself; but the loud, brash proclamations of the Communist Manifesto- i.e. "workers of the world unite, you have nothing to lose but your chains"- have cemented themselves into our culture much better than his later writings, some of which weren't even published until the 1930s).

As an example, many young Hegelians blamed cultural values (like religion) for many of the woes of the Germany they lived in. Marx affirmed the existence of these same inequities but argued they proceed from the economic circumstances of Germany. When Marx said "religion is the opiate of the masses," this wasn't strictly a condemnation of religion- as you can tell if you look at the full quote:

The foundation of irreligious criticism is: Man makes religion, religion does not make man. Religion is, indeed, the self-consciousness and self-esteem of man who has either not yet won through to himself, or has already lost himself again. But man is no abstract being squatting outside the world. Man is the world of man – state, society. This state and this society produce religion, which is an inverted consciousness of the world, because they are an inverted world. Religion is the general theory of this world, its encyclopaedic compendium, its logic in popular form, its spiritual point d’honneur, its enthusiasm, its moral sanction, its solemn complement, and its universal basis of consolation and justification. It is the fantastic realization of the human essence since the human essence has not acquired any true reality. The struggle against religion is, therefore, indirectly the struggle against that world whose spiritual aroma is religion.

Religious suffering is, at one and the same time, the expression of real suffering and a protest against real suffering. Religion is the sigh of the oppressed creature, the heart of a heartless world, and the soul of soulless conditions. It is the opium of the people.

The abolition of religion as the illusory happiness of the people is the demand for their real happiness. To call on them to give up their illusions about their condition is to call on them to give up a condition that requires illusions. The criticism of religion is, therefore, in embryo, the criticism of that vale of tears of which religion is the halo.

Marx adds depth to the long standing Enlightenment critiques of religion by pointing out that religion was the product of socioeconomic circumstances, and to renounce oppressive religious constructs necessitated renouncing oppressive economic relationships (serfdom, slavery, wage labor, etc.) that gave birth to religion in the first place.

Consequently diamat leads to the conclusion that ideas (and systems of ideas aka ideologies) are themselves influenced by the material base of society, rather than the other way around as Hegel argued (Marx on Hegelian dialectics: "With him it is standing on its head. It must be turned right side up again, if you would discover the rational kernel within the mystical shell").

The material base refers to the economy, the socioeconomic relationships between people. This is a simplification, but the basic idea is that in a capitalist society, ideas which benefit capitalists tend to propagate and be favored over those that don't benefit them or which actually hurt them. In The German Ideology Marx argues:

The ideas of the ruling class are in every epoch the ruling ideas, i.e. the class which is the ruling material force of society, is at the same time its ruling intellectual force. The class which has the means of material production at its disposal, has control at the same time over the means of mental production, so that thereby, generally speaking, the ideas of those who lack the means of mental production are subject to it. The ruling ideas are nothing more than the ideal expression of the dominant material relationships, the dominant material relationships grasped as ideas; hence of the relationships which make the one class the ruling one, therefore, the ideas of its dominance. The individuals composing the ruling class possess among other things consciousness, and therefore think. Insofar, therefore, as they rule as a class and determine the extent and compass of an epoch, it is self-evident that they do this in its whole range, hence among other things rule also as thinkers, as producers of ideas, and regulate the production and distribution of the ideas of their age: thus their ideas are the ruling ideas of the epoch. For instance, in an age and in a country where royal power, aristocracy, and bourgeoisie are contending for mastery and where, therefore, mastery is shared, the doctrine of the separation of powers proves to be the dominant idea and is expressed as an “eternal law.”

(emphasis added)

Marx, being a historian, formulated this idea specifically with respect to history and historiography but Marxists have since applied the idea elsewhere.

So, to answer your question, Lysenko believed that the Darwinism which undergirded Western eugenics (not just in Nazi Germany, but in the US, UK, France etc. also) was the result of bourgeoisie material conditions. This is quite correct, I would argue. His mistake was then throwing evidence based science out the window and attempting to fit the data to the ideology, while not realizing Darwinian evolution and Mendelian inheritance does not in any way imply that eugenics is a good thing. It would be unsatisfactory to reject a Marxist philosophy of history simply because of the failure of Lysenkoism.

The discussion around Lysenkoism often makes it seem uniquely bad in history, and people often ascribe bizarre death tolls to it with little evidence. In truth, ideologically guided science is not the historical exception; it is the rule. Western science at the time was just as ideologically guided by the dominant material realities of capitalism and racism, which resulted in widespread support for eugenics and "race science" intended to justify slavery and imperialism, as well as scientific human experimentation on subjects deemed subhuman (Tuskegee syphilis studies and Dr. Mengelev's experiments, for example; if you're interested in learning more, the book Medical Apartheid is a good book on the scientific experimentation on black Americans, and Gould's The Mismeasure of Man is a good book on race science). These had a human cost that was certainly higher than Lysenkoism.

Part 1:

/u/specterofsandersism has explained Marxist views on the dialectic very well, but Trofim Lysenko's motivations and influence on Soviet science were not just pure ideology, but instead need to be put in historical and scientific context. The Soviets weren't ideological robots, and their motivations for pushing theories were more complex than what Marx said, taking in issues of Russian nationalism and being responses to particular problems. It's also been argued that Lysenkoism (as its Western detractors called it) or Michurinist biology (as its supporters called it) was the crucible in which modern Western views of pseudoscience were formed; as a result, Lysenko's motives and effects have been hotly contested, with different people on different sides of arguments pushing their own barrows about the extent to which Lysenko was scientific, depending on what they think about the idea of pseudoscience.

To sum up the facts very briefly, Lysenko became prominent in Soviet science in the late 1920s for his work on what's now in English called 'vernalization' - basically inducing winter crops to grow in summer by applying moisture and cold to the seeds. This led him to become prominent in Soviet scientific circles, and as the 1930s progressed, he came to strongly push the barrow of a basically Lamarckian view of evolution, arguing in a semi-mystical way that living things could inherit characteristics of their parents - the famous Lamarckian example being a life of giraffes striving to reach the leaves on trees having an effect on the length of their childrens' necks (as opposed to Darwinian views of evolution which see the unit of inheritance being genes, and mutations as the only way for evolution to occur).

Note that, before the discovery of DNA by Watson and Crick (not to mention Franklin) in the 1950s, Lamarckian thought wasn't as implausible as it now seems today, because the unit of inheritance was something of a mystery - Mendelian genetics wasn't as convincing as it is today (especially considering that simple Mendelian genetics can only be demonstrated easily with certain somewhat 'on-off' phenotypical traits of individuals - eye colour, for example - most traits are instead the result of a melange of different genes). And note that epigenetics means that the lives of ones grandparents can effect the way that genes are expressed - see this example - and so there was some limited support for Lamarckian inheritance theories that Lysenko heavily overextended.

In general, Lysenko is usually contrasted with Nikolai Vavilov, his big rival for prominence in Soviet biology in 1930s, and - you guessed it - an adherent of Western-style genetics. Because of the severe famines that the USSR had experienced, and their limited resources, agricultural science was heavily politicised in the USSR, to the point where Stalin and Molotov were meeting with Vavilov and Lysenko in order to roar at them to improve harvests. Gary Paul Nabhan's Where Our Food Comes From: Tracing Nikolay Vavilov's Quest To End Famine is much more sympathetic to Vavilov than Lysenko, claiming that, in these meetings, Lysenko peddled quick cheap fixes that didn't actually solve a lot of the problems in harvests (i.e., lots of vernalisation), which stopped Vavilov from being able to make the long-term changes that would have actually increased yields significantly.

Vavilov lost political ground to Lysenko through the 1930s for a variety of reasons that might go beyond ideology; at one point Stalin shouted 'GO AND LEARN FROM THE SHOCK-WORKERS IN THE FIELDS!' at Vavilov when he suggested going overseas to learn new crop yield techniques. Vavilov also antagonised Stalin and Molotov by pointing out in public that crop yields were higher before the revolution, and it seems that the hierarchy thought Lysenko's quick fixes were more achievable. In the end, Vavilov was arrested in 1940 and died of starvation in prison in 1943.

The Soviet famine of 1946-47 sadly appeared to show that Lysenko's Michurinist biology was not any more successful at increasing yields than previous agricultural techniques (the widespread application of vernalisation, seemingly, had the effect of increasing short-term gains but depleting the soil). Nonetheless, in 1948, Lysenko had risen to become the President of the Lenin All-Union Academy of Agricultural Sciences, and at their conference on July 31 he announced that modern biology had diverged into two opposing trends - a Soviet science called 'agrobiology' or 'Michurinist biology' based around Lamarckism and a Western science which Lysenko called 'formal' genetics, which was based around Mendellian genetics ('Mendelism-Morganism-Weismannism'). Lysenko prominently argued here that 'formal' genetics was 'bourgeois'.

He had been arguing this for 15 years by this point, and was already one of the main voices in official policy, but what changed in 1948 was that Pravda heavily publicised this argument. Suddenly this became a hot-button issue; newspapers published articles glorifying Lysenko, and hundreds of thousands of copies of the proceedings of the conference were published. By the end of 1948, Lysenkoism was official policy. Genetics laboratories run under the principles of 'formal' genetics were closed, biologists fired, and courses at universities abolished. Mendelian genetics had officially been banned in the USSR and its satellites; in December 1948 a movie about Lysenko was released directed by a famous director, Aleksandr Dovzhenko, with a score by Shostakovich, and with one of the most popular actors of the time, Grigorii Belov, in the title role.

/u/specterofsandersism is correct that the link between 'formal' genetics, eugenics, and fascist thought was pushed heavily in Soviet literature in this 1948 period as a justification for Lysenkoism. Lysenko had consistently been opposed to eugenics since the 1920s, and 1948 was a period when the horrors of eugenics were clearer, post-Nazi Germany, than they had been in previous eras. However, before Lysenko ascended to prominence, there had been various attempts at creating Bolshevik/Soviet eugenic programs which were intended to improve society as a whole. In the post-war period, however, the Soviets could use their scientific opposition to eugenics as an effective propaganda point in the nascent Cold War. This was a point in time when there was a fair bit of sympathy for the Soviets in Western intellectual circles, before the 1956 Hungarian revolution was crushed, and before the full extent of Stalin's brutality was clear. The USSR being the kind of ethical place that had the empathy for humanity to be against eugenics still seemed plausible in such circles at this point, however, and so it made for good propaganda material.

Lysenkoism lasted as official policy until 1962; Khrushchev ended it as part of his de-Stalinisation policy. 1962 was close to a decade after DNA had been discovered, making Lysenkoism much harder to justify. Thanks in part to the application of an increased understanding of genetics, corn grain yields in the USA, for example, doubled between 1940 and 1960 (they also came close to doubling between 1960 and 1980, meaning that they had almost quadrupled in 40 years). Because Lysenkoism was official policy in the USSR until 1962, it was only in the 1960s when the Russians seem to have been able to effectively predict crop yields.

It's fair to say that Lysenkoism didn't do a great deal to help in the 1946-1947 Soviet famine, and that alternative approaches that were available at the time would have helped more; Lysenko's policies had a tendency to increase yields in the short-term but decrease them in the long-term, and Lysenko discouraged the use of American corn breeds with increased yields because he didn't believe they worked. I'm not quite clear in my reading about the extent to which the 1948 purge was basically finding scapegoats for the famine, but I strongly suspect that 'bourgeois' genetics was a convenient scapegoat for the famine at this point, much as Vavilov became the scapegoat in the 1930s (and that there's a big dose of Russian nationalism that's an inherent justification of home-town hero Lysenkoism).

In science, you need theories to explain data, and the theories you choose and the data you choose to emphasise are influenced by your assumptions about the world, which are ideologies to a lesser or greater extent. It's not enormously surprising that Lysenko as a committed Stalinist had a view of the world coloured by that ideology, and that his science was therefore also coloured - this is the nature of science. In the high-stakes world of Stalinist politics - which interacted with agricultural science because of famines - it's also not surprising that Lysenko made power plays based on trying to appeal to Marxist ideology as interpreted by the USSR at the time, and that he was quite successful doing so.