It didn't. The central (though far from only) divergence of Leninism from classical Marxism to allow it to work in early 20th Century Russia, which wasn't the kind of advanced (even by mid-late 19th Century standards) liberal industrial capitalist regime to which the Communist Manifesto was directed, was the adoption of vanguardism, in which broad class consciousness and awareness of class theory by the proletariat was abandoned as a requirement for Communist revolution in favor of authoritarian leadership of the masses by a narrow intellectual elite.
A major justification for Leninism was the belief by Russian Communists of the time that what you describe could not happen in Russia.
(I don't assert that the reason for that is or is not anything to do with culture of reading, but the Communist Revolution certainly does not support your thesis in the way you suggest.)
Firstly, the leaders whom the Bolsheviks overthrew in October 1917 were not familiar to most Russians. The Provisional Government instituted after the February Revolution was a messy affair.
Committed Bolsheviks were a minority at the time of the Russian Revolution. However, there were just enough of them in the railroads, ports, army and other key infrastructure that they could carry out a coup against the Provisional Government.
As with many radical movements, most of the ordinary Bolshevik sympathizers had never actually read the foundational texts of Communism (Marx' writings are often a challenge even for intellectuals). They only knew some simple agitprop distillations thereof.
That the Communist regime spent the whole 1920s and early 1930s carrying out literacy campaigns and establishing written standards for most of Russia's languages also underscores how little of a culture of reading existed in the country pre-1917 outside the elite cities like Moscow and Saint-Petersburg.
It didn't and couldn't. That was the conceit and power behind Lenin's "vanguard" concept. Only a small minority needed to be ideologically driven; everybody else could and did join sides based on more practical, individualized motivations. See, generally, https://en.wikipedia.org/wiki/Vanguardism
That said, I'm not trying to opine on whether, when, and how popular book reading was or still is.
not disagreeing with your premise, but -
assuming this to be accurate (leaving out alot of factors),
nothing says small number of people couldn't read it and then go around verbally agitating everyone else.
that said literacy was fairly high in those times and i agree reading was very common
How can you say that? Literacy was once a very strong proxy to wealth, and still retains a connection today. Books have been fundamental to many aspects of human history.
If Russia never had a culture of reading, then how does the Communist Manifesto drive an entire society, millions of people, to overthrow its familial leaders before the era of television?