Five people were stabbed to death and another died after jumping out of a window to escape the blaze, police said.
Seven others were wounded, including four seriously, and the death toll could rise, according to police.
The suspect, arrested at the scene, told police he did not want to live because "everybody looks down on me," Kim Kap-shik, chief detective at Seoul's Gangnam Police Station, told reporters.
Police also seized two more knives and a tear gas gun from Jeong.
The suspect used to work part-time at restaurants and other places, but has been out of job since April, police said.
Jeong's "livelihood was difficult and (he) has been under considerable financial pressure" and could not pay his rent and mobile phone fees for months, the investigator told reporters.
Kim also said Jeong was facing a police investigation for not taking part in annual training for military reservists.
Jeong also told police he attempted to commit suicide when he was a middle school student, and had been suffering occasionally from severe headaches.
Yonhap news agency and other media reported that Jeong has been convicted of crimes eight times in the past. Police were not immediately available to confirm Jeong's criminal record.
One of the dead and four injured victims were ethnic Korean Chinese citizens working in South Korea, according to media reports.
Such random violence is not common in South Korea, though is not unknown.
In February, a 69-year-old man, upset over a land dispute, started a blaze that destroyed a 14th-century gate in Seoul that is considered one of South Korea's most treasured landmarks.
And in 2003, a 56-year-old man with a record of mental illness ignited a carton filled with gasoline on a subway train in the southern city of Daegu, leaving 198 people dead and 147 injured.
Neighboring Japan has been struck by a series of random stabbings this year.
In the worst case, seven people were killed in Tokyo's Akihabara electronics district in June when a man slammed a truck into a crowd of people, jumped out and began stabbing passers-by at random.