Koznyshev (the famous writer) is actually Levin's half-brother.
Levin has been staying with Koznyshev while in Moscow, and after he gets home, he wanted to ask his brother's advice about proposing to Kitty.
Koznyshev is having a debate with a philosophy professor over whether or not there's a difference between psychic and physiological phenomena in human behavior – and if yes, where that split is.
In other words, Koznyshev disagrees with the idea that everything we can know of the world comes from our senses. He seems to be arguing for some kind of innate, preexisting notion of being.
The professor, on the other hand, argues that consciousness is the result of your sensory impressions. In other words, he argues that who you are emerges as a result of what you see, touch, taste, and so on.
Koznyshev acknowledges his brother, introduces him to the professor, and then the two continue their debate.
Levin, listening to this debate, is frustrated: he feels that they are both missing the point. He asks: if our bodies die, then, is there nothing left of us? Levin is asking about the immortality of the soul, which neither Koznyshev or the professor seem to care about.
The professor dismisses Levin's question as something that can never be known.