Nadsat--or teenage--slang sets Alex and his friends off as a subculture. They don't speak standard English, and this helps establish that they look at the world through a different lens. Further, this language continually reminds us that we are in a different, future world that is not our world, reinforcing the science fiction aspects of this novel.
The slang is deliberately disorienting, and it forces us to slow down and really read the words on the page, rather than sliding over them glibly. We have to concentrate and think about what is being said in order to understand it. Often Alex and his friends use a slang term that brings a different connotation to a word. For example, instead of "cigarette," a fairly neutral term, they use "cancer," a much more loaded word that reminds us of the harshest implications of cigarette smoking. This also shows that they see through the pretenses of a world that pretends it is not out to destroy them.
Burgess, who wrote a book called ReJoyce about James Joyce, owes a debt to Joyce's use of vivid slang and invented words. Words such as "barry place" for prison, which play on the idea of prisons as full of bars--bars on windows, bars on the doors of cells--brings up visual images of what being incarcerated is like in a way the word "prison" does not.
The slang is integral to establishing the alternative universe of the novel, a world of teens who do not necessarily want to invite us in.