That story is set in Berlin, the informal capital of Russian émigrés, where Nabokov lived from 1922 to 1937. His father had set up the émigré newspaper Rul', in which Nabokov - writing under the pseudonym V. Sirin - also published his first poems and novels. The father was assassinated in 1922 by a Russian monarchist, but Nabokov continued living in the city, earning a small income as a private tutor of languages and tennis.
The Eye is one of Nabokov's two published novellas (the other one is Transparent Things from 1972) and was his fourth book, written in Russian. The English translation was made in 1965 by Nabokov's son, Dmitri.
When I looked at reader's comments on the internet, I was struck by the large number of people claiming not to understand this novella. Indeed, there are two hurdles, an easy one and a harder one. To solve them, it is important to note that throughout his career, Nabokov invented many tricks such as literary puzzles, plot intricacies and unreliable narrators to manipulate the reader, but that he also always adhered to the rule of fair play - the reader is given sufficient clues to work out what is going on.
The easy hurdle is of course that the (unnamed) narrator commits suicide early on in the story. He has been beaten and humiliated in front of his two pupils (who keep staring at the scene with big eyes) by the jealous husband of a woman he was bedding. Out of shame he later shoots himself in his room. But Nabokov gives plenty of hints that the suicide is unsuccessful (although at the same time preserving a decided ambiguity): the narrator feels uncomfortable with the pistol pushing against his chest, so he moves it away before pulling the trigger; and he wakes up in a hospital. Still, after the shot has gone off, he believes himself to be dead and now a ghost walking the earth. I would call this a psychological condition rather than reality, for even in Nabokov ghosts don't exist. Anyway, he continues to go about his daily activities as if he had never died.
We then get an ironical story about the Russian émigré milieu, a parody on the all-Russian theme of a neurotic and tortured character, who fails to make an impression on the (higher class) people with whom he associates, and who in the end brings about his own humiliation. It is good to realize that Nabokov strongly disapproved of this cult of suffering and self-negation, and that he hated Dostoyevsky's 'humiliated heroes.' Nabokov had a very low opinion of the sentimental Dostoevsky, so he makes fun of him in this brilliant parody.
This Smurov is an enigmatic young man, flatteringly introduced with the words 'everything he said was intelligent and appropriate,' but who also likes to brag about his imaginary exploits fighting the Communists during the Russian revolution - unconsciously revealing that he is a liar, for when he relates a heroic feat taking place at the Yalta railway station, another visitor coolly remarks that there is no railway station in Yalta. Of course, the bragging serves to attract the attention of a young woman, Varvara, nicknamed Vanya, with whom he is in love. He lovingly describes Vanya with the words 'she was so enchanting... her downy face, near-sighted eyes... her short bright dresses... her big knees and big hands with pink knuckles.' (Not coincidentally, she has a masculine nickname.)
On top of that, the lovesick Smurov is the only one who does not know that Vanya is in fact already engaged to another visitor, Mukhin.
In this comic-grotesque tale the narrator steals other people's correspondence to get their opinion about himself, and he also sneaks into Vanya's room to see if she has his picture on her desk - only to discover that his image has been carefully cut away from all pictures she keeps! Now ends Smurov's obsessive attempt at self-detachment and he passionately declares his love to Vanya - and is sent packing with his tail between his legs.
At the end of the story, the narrator again happens to meet the irate husband from the beginning - who now is the first to call him "Mr Smurov," thereby confirming the single identity of these two pseudo-doubles - and is offered an apology as other infidelities of the wife have come out. The narrator then closes with a desperate, strident protest of happiness. I suspect that the narrator/Smurov is again unreliable here, and that this flattering meeting with the husband is just a fantasy.
We could call Smurov a spy (the meaning of the Russian title) of what other people think about him. He tries to find his identity in the mirrors of other people's opinions, something which is of course impossible, for others may have completely wrong ideas about us. Identity is not something that can be spied out, but a reality that must exist in ourselves.