South Koreans arrive in Dubai amid claims a ransom was paid for their freedom.
Yu, fighting back tears, also expressed condolences to the families of two colleagues who were killed shortly after the kidnapping on July 19.
Their relatives, holding photos of their lost loved ones, also appeared with the hostages at a brief news conference.
“They looked extremely subdued, very quiet, some of them were holding hands, one of them was crying. They looked totally overcome by the situation,” Tony Birtley, Al Jazeera’s correspondent in Seoul, said.
The hostages were then reunited with their relatives, who cheered as the group entered the room in a hospital near the airport.
They will undergo medical checks and a debriefing from the South Korean foreign ministry.
The Taliban on Saturday denied a claim by one of its own commanders that the group received $20m from Seoul in return for freeing the South Korean missionaries.
One Taliban commander earlier told the Reuters news agency the money had been received and would be used to “purchase arms, get our communication network renewed and buy vehicles for carrying out more suicide attacks”.
The source, reportedly on the 10-man leadership council of the group, said: “We got more than $20m from them [the Seoul government].
“The money will also address to some extent the financial difficulties we have had.”
But Youssef Ahmadi, a Taliban spokesman, told Al Jazeera that the group had not received any money and claimed the reports were part of a campaign to discredit the Taliban.
Alan Fisher, Al Jazeera’s correspondent in Kabul, said: “It is in the interests of both the Afghan government and the Taliban to deny reports of a ransom.
“But Al Jazeera has been told by more than one source on more than one occasion, one of them a senior figure in the Taliban, that a ransom, said to be around $20m, has been paid.”
Some of the 19 released hostages have told Al Jazeera in a series of exclusive interviews how they lived in constant fear for their lives during their captivity.