It's left pretty obvious for the most part - the writer handles it well and credits his reader with some sense, letting them work it out for themselves - gradually, in some instances.
Looking at the first few lines:
What's it going to be then, eh?
There was me, that is Alex, and my three droogs, that is Pete,
Georgie, and Dim, Dim being really dim, and we sat in the Korova
Milkbar making up our rassoodocks what to do with the evening, a flip
dark chill winter bastard though dry.
Droogs and rassodocks explain themselves in the context, more or less.
Rassoodock must mean 'mind' - making up our minds is the most obvious way to complete the sentence in a meaningful way.
Droog is less sure - at first glance it could be 'friend,' 'boy,' or anything of that sort - but we have enough to proceed, and the meaning can be clarified as we go along. (This is not at all dissimilar to how children pick up words.)
It's worth noting, too, that not all Burgess's Nadsat is Russian in origin, or even Slavic: ultraviolence or cutter (money), for instance. The reader is expected to figure out those as well (in the same manner that anyone who has read this far but had not previously encountered the word Nadsat would now have a very good idea of what it means, without ever having explicitly been told).
As to Burgess's specific motivation for creating and employing Nadsat, Wikipedia says:
Burgess, a polyglot who loved language in all its forms, was aware
that linguistic slang was of a constantly changing nature. Burgess
knew that if he used modes of speech that were contemporarily in use,
the novel would very quickly become dated. His use of Nadsat was
essentially pragmatic; he needed his narrator to have a unique voice
that would remain ageless while reinforcing Alex's indifference to his
society's norms, and to suggest that youth subculture existed
independently of the rest of society.
And it worked - Nadsat has helped put A Clockwork Orange and its setting outside any specific time. I don't doubt that's part of the book's enduring appeal, which is the primary reason why we are discussing it even now.
The little chunks of Russian and all the aberrant speech of Alex and his droogs are by no means unique as a literary device, either - fantasy and sci-fi authors often establish an idiom of this kind, for reasons of style, to stress the alien not-here-and-now-ness of the setting or to introduce necessary vocabulary for new concepts, with the wholesale creation of complete languages with their own writing systems, such as Klingon and Elvish, lying at the extreme end of this phenomenon.
User GEdgar noted in the comments above that there was a glossary in the paperback edition he read. I don't remember one in the imprint I read, although I could very well be wrong. Nonetheless, this has a direct bearing on the correct answer if how readers actually understand the Russian vocabulary is simply, 'They look up the words in the back of the book.'
But, at least for some readers, GEdgar perhaps among them, this is definitely part of the answer.
In the first edition of the book, no key was provided, and the reader was left to interpret the meaning from the context.
This is a frustratingly less-than-complete answer (and I wasn't reading a first edition!)
Looking at the introduction to the Penguin Modern Classics edition, written by Blake Morrison, and available to read through Amazon's 'Look Inside' feature:
The old American edition of A Clockwork Orange carried a glossary of nadsat words. Burgess did not approve of this: in a novel which takes brainwashing as its subject, he intended his own form of brainwashing, which was to force readers to use a Russian dictionary.
We can at least now see that not all imprints carried a glossary, and that there may be a British/American split on this.
I feel Blake Morrison is being a little flippant when he suggests that Burgess wanted his readers to use a Russian dictionary, as he immediately goes on to say:
Though reading the novel requires some puzzle-solving, the meaning of a nadsat word is often clear from the context, or from Burgess's own glosses: a paragraph about the pleasures of deng obligingly ends: 'But, as they say, money isn't everything.'
So the answer to how readers understand Nadsat is: context, sometimes a glossary at the back of the book, and these days, most likely, by looking them up on the net when they encounter one they struggle with.
(For the curious, the text of the Nadsat glossary from the first American edition can be found here.)