For myself, as an English speaker with only a smattering of Russian from a foreign exchange trip, I did a lot of checking the glossary in the back at first (obviously, I got an edition which included said glossary), but eventually started just trying to figure it out from context, which worked well enough.
As regards Burgess's motivation, it was a combination of his fascination with language, a desire to avoid the novel being dated by picking a particular era's slang, and use of language to show a counter-culture, as seen in this article:
Much like George Orwell with his ‘Newspeak’ in Nineteen-Eighty Four (1949), Burgess aimed to create a timeless language to depict his dystopian future, perhaps the reason why the novel has had such longevity. The language also removes the action of the novel from geographical location, and the city it is set in could stand for anywhere from Manchester to Leningrad, London to Los Angeles, or other even more distant locales.
Burgess viewed his use of Nadsat as a ‘brainwashing device’, something he writes about in You’ve Had Your Time (1990): ‘The novel was to be an exercise in linguistic programming, with the exoticisms gradually clarified by context: I would resist to the limit any publisher’s demand that a glossary be provided. A glossary would disrupt the programme and nullify the brainwashing’. The book’s editor, James Michie, had some hesitations about the density of Nadsat in the novel and stated a desire for Burgess to ‘make it gently accelerando. You can’t throw too much of it at them too quickly’. This editorial suggestion led to revisions in the first part of the novel, and is shown when Alex helps the reader through some of the tougher language. For example: ‘rooker (a hand, that is)’, ‘litso (face, that is) and ‘my three droogs, that is Pete, Georgie, and Dim’.