All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
Indeed Allaah has made permissible for us to eat the meat of animals slaughtered by the people of the book. However, the conditions that make the meat permissible for us to eat are the same regardless of whether the animal was slaughtered by the people of the book or by Muslims. Under all circumstances, the animal should be slaughtered according to known Islamic rites, and not be killed by electrocution, hanged to death or hit by a heavy object. It is also a condition that this animal not be slaughtered for other than Allaah (mentioning other than the name of Allaah upon slaughtering it). This issue is discussed in details in Fataawa 82899 and 82323.
As regards the quote the questioner is attributing to Shaykh Al-Qaradaawi, we have no knowledge of it; therefore he can refer to the people running his website on this matter.
Allaah Knows best.