Symbolism, Imagery, Allegory
Andrei's bizarre relationship with his cats embodies the twisted path his life has taken.
Although Andrei has a family, they're secondary to his feline friends. He hardly sees his kids. He ensures that "his wife didn't attempt to hug or kiss him" because she knew "he was strict about these things" (4.35.28). Heck, every moment that he isn't on a work trip (er, murder spree) is spent in his basement with his cats.
His family is well aware of this odd behavior. His daughter Nadya, for example, spends her days "wondering what kind of father preferred cats to children" (4.35.38). At times, we even see a bit of her dad's insanity in her when she plots the demise of her dad's cats.
Ultimately, this relationship reflects Andrei's detachment from the human race. The guy feels like he's been let down by everyone in his life, from his mother Oksana to his brother Pavel. With that, his only salvation becomes his cats—symbols of the childhood experience that made him the man he is today.
The cats are also safe for him. In this society, where everyone is under suspicion, all the time, you can't ever trust anyone, not even members of your own family. Cats can't tell on you, or turn you in to the authorities; you can have affection for them without getting into trouble. (And you don't have to kill them afterwards.) It's totally messed up.