There are basically two scholarly takes on why Lysenkoism was supported. The first one is simultaneously old and new; it is the take that people said about Lysenkoism at the time, but has gotten "refreshed" by some modern historians of science as well. It basically says, "because Lysenkoism was more ideologically compatible with Marxist-Leninist-Stalinism than Mendelism." This is the idea that the Soviet government support Lysenkoism because it preached the mutability of biology and thus promised infinite malleability for Soviet man and his endeavors. So this is what anti-Soviet scientists (often stimulated by covert CIA funding) argued during the Cold War, when they made Lysenkoism into one of the many reasons to oppose Stalinism.

The other take, which was especially common in scholarship from the 1970s until very recently, is that ideology had nothing much to do with it. Rather, this interpretation says that Lysenkoism became initially powerful because the Soviet system created a political environment that ended up rewarding any "sciences" that could simultaneously claim practical results (however spurious), be used as domestic propaganda, and effectively denounced its opponents as being associated with the West and being ineffective. Lysenko was an effective political operator, in this view, and provided the Soviet state with a ready-made way to deflect criticisms of its agricultural policies, and the inability of early Mendelian geneticists to translate their theoretical work into practical results led to easy attacks. (Hence they were "fly-lovers and people-haters" in Lysenko's argument.) Lysenko made for good domestic propaganda, and that made him powerful; Stalin found him useful, and thus elevated him to the favored position.

So which of these is right? Well, they can both be sort of right. The modern "Lysenkoism as ideology" scholarship emphasizes Stalin's own ideological issues as being the reason why he decided to elevate him into the favored position, but the other elements probably better explain why Lysenkoism was so successful in the particular political situation of the 1930s USSR. Appealing to Lysenkoism as just being about Soviet ideology seems to misunderstand how ideology and power work in totalitarian nations (and notably the USSR), seeing ideology as sort of a given rather than something that is "worked out" by the actions of the state (and not just a "top down" state either; much of the campaigns for or against various Soviet sciences were waged by academics vying for power and security, not top-level Party officials). But once Lysenko became an important force, Stalin's own whims — which could be ideologically influenced, and he considered himself quite the philosopher and scientist — could lead to such a situation being codified.

As for the consequences: Soviet biology became highly politicized and until some time into the Khrushchev years, Mendelian genetics was essentially verboten, and many Mendelians ended up in the Gulag, at times fatally. Lysenkoism was essentially scientifically empty; its only "success" was its vagueness, which made it hard to be falsified. There are some arguments that Lysenkoism was complicit in some ways in the Soviet famines of the period though I suspect this is a bit of a stretch (collectivization was disastrous no matter what biological theory one had in mind). Even after Khrushchev dethroned Lysenko, Lysenkoism persisted for some years in the People's Republic of China as well.

On the use of anti-Lysenkoism in CIA-funded scientific propaganda, see Audra Wolfe's Freedom's Laboratory. On the "non-ideological" explanation of Lysenkoism, the classic work is David Joravsky's The Lysenko Affair. On the modern "re-ideological" discussions, see Nils Roll-Hansen's The Lysenko Effect.