The setting for about half of the novel The Kite Runner, by Khaled Hosseini, is Kabul, Afghanistan. Amir, the protagonist and narrator of the story, lived in Kabul as a boy and remembers the city fondly, other than some terrible choices he made in his personal life. WHen the Taliban arrives, Amir and his father are forced to flee Kabul for political reasons.
Amir and his father leave behind two important people: Hassan and Rahim Khan. Hassan is Amir's half-brother and Baba's illegitimate son; Rahim Khan is Baba's business partner and a family friend. Rahim Khan promises Baba to stay in the house and maintain it, but the job gets to be too much for him and Hassan sacrifices the safety of living quietly in an isolated village with his wife for doing his duty to his family.
When Rahim Khan sends for Amir and tells Amir he has a chance to make right some of the great wrongs in his own and in Baba's life, Amir goes to see Rahim Khan who is now living in Pakistan. He gives Amir a letter Hassan wrote to him but has to tell Amir that Hassan is now dead.
The letter kind of fills in all the gaps about Hassan's life since Amir and Baba left Kabul, and it contains the line you mention. Since Amir has not been back to Kabul for nearly twenty years, he does not know what happened there after the arrival of the Taliban. Hassan tells him:
The infighting between the factions was fierce and no one knew if they would live to see the end of the day. Our ears became accustomed to the whistle of falling shells, to the rumble of gunfire, our eyes familiar with the sight of men digging bodies out of piles of rubble. Kabul in those days, Amir jan, was as close as you could get to that proverbial hell on earth. Allah was kind to us, though. The Wazir Akbar Khan area was not attacked as much, so we did not have it as bad as some of the other neighborhoods.
The letter continues, giving more details about Hassan's young son and the trials of living in Kabul. The quote you mention (which I made bold, above) is a succinct summary of the horrors Hassan and others felt while living under Taliban rule in Kabul, Afghanistan.
The letter does what Rahim Khan hoped it would, and Amir decides to go to Kabul and rescue Hassan's son.
When Rahim Khan relays to Amir what Kabul was like at the end of the fighting with Russia and the start of the Taliban rise to power, he tries to give a picture of what Hassan and his family faced. At one point in time, Kabul used to be a centre of activity and a bustling metropolis. Even amidst the fighting, Amir's memory of Kabul was one in which there was total and complete activity. What Kabul became in his absence was a "proverbial hell on earth." The forced impositions of life from the Taliban as well as the recovery from years of fighting with the Soviets transformed a city teeming with life to a vision of the underworld in which destruction, suffering, and death were present.
This depiction of Kabul has to be as visceral as possible in order for Amir to have a full grasp of both what happened to the city he once knew and what he is going to see in his journey within it. The meaning of the quote helps to illuminate this condition. It is one in which what was has been transformed into what should never be. In being able to render such a portrait, it becomes clear that the rich history of Kabul, and Afghanistan, has been permanently transformed through years of war and violence. The description of Kabul as a "proverbial hell on earth" brings this out to Amir and the reader.