At the time, he thought the heart was intended for a critically wounded KLA officer, Serbian news agency Tanjug reported.
The unnamed man, a witness in a war crimes investigation, made the claims Monday on Serbian public television network RTS. His image was obscured during the broadcast.
War crimes prosecutors are investigating allegations that in 1999 and 2000 members of the KLA removed the vital organs of kidnapped Serbs and non-Albanians and sold them on the international black market.
Deputy War Crimes Prosecutor Bruno Vekaric said the man had agreed to cooperate with the prosecution.
"Training was done in northern Albania for various activities, including for medical interventions," Vekaric said. "He was told there that he was being trained because a KLA commander could be injured in the front, so he should learn how to transplant a human organ. He was not told this was a criminal act of organ extraction."
In the broadcast, the witness described a 19- or 20-year-old man being held down on a bench by four men. The witness said a doctor handed him a scalpel and told him to remove the man's heart.
"After I cut him, at some point half way, he started screaming for us not to kill him, and then he lost consciousness, I don't know if he fainted or died...," the witness said.
He said the heart was placed in a box, which he accompanied to the airport in Tirana. There, the box containing the heart was given to man in a plane displaying the Turkish flag.
"I realized this was not about the fatherland, but about the business of organ trafficking, about selling. It is not about ideals but about something ugly," the witness said.
The prosecutor said afterward that the televised statement was "only 1 percent of everything he (the witness) had to say."
Vekaric said officials from other war crimes courts would be able to interview the witness to verify his credibility.