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.
An official in South Korea's presidential office also said on Saturday:"We deny any payment for the release of South Korean hostages."
"The two conditions for the release are that we pull out our troops and stop Korean missionary work in Afghanistan by the end of the year," said the official, who declined to be named.
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."
Fearing for their lives
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.
They said they were split up into small groups and moved around the Afghan countryside to avoid detection.
One Taliban member would tend to a farm by day and then grab a rifle and stand guard over hostages at night, they said.
Television footage of the freed hostages broadcast on Friday raised concerns about their health and the possible effect media exposure could have on their fragile emotional state, Cha Seong-min, who has represented family members since the crisis began, said.
"I'm very much concerned because she looked like she lost a lot of weight," Cha said of his sister, Hye-jin, one of the 23 South Koreans captured.
The kidnapping was the largest in the resurgent Taliban campaign against foreign forces since US-led troops removed the Islamists from power in 2001.
The Taliban decided to free the hostages after Seoul restated its plan to pull all its forces out of the central Asian country.
Seoul had already decided before the crisis to pull its 200 engineers and medical staff out of Afghanistan by the end of this year.
It has also banned nationals from travelling there.
Source: Al Jazeera and agencies