This is well known, for if we were to interpret “their food” here as bread, dates, and the like, there would be no distinction between the People of the Book and those who are not from the People of the Book. For it is likewise lawful for us to eat the bread, dates, and similar foods of those who are not from the People of the Book. What is meant by their food, then, is their slaughtered animals. It was attributed to them because they slaughtered them in order to eat them.
It is also established in the sunnah that the Prophet, peace and blessings be upon him, ate animals slaughtered by the Jews. A woman gave him a sheep as a gift at Khaybar, and he accepted it (1). A Jew invited him to a meal of barley bread and rancid fat—that is, fat that had changed and developed a foul smell—and he accepted (2) and ate. It is likewise authentically established that Abdullah ibn Mughaffal, may Allah be pleased with him, took hold of a leather bag of fat that had been thrown down at Khaybar. He turned around and found the Prophet, peace and blessings be upon him, behind him, laughing (3). This is an instance of tacit approval in the sunnah.
If you say: Is it not possible that the one who slaughtered it was a Muslim?
The answer is: This is a remote possibility and contrary to what is apparent. Had what they presented to the Messenger, peace and blessings be upon him, not been lawful unless it had been slaughtered by a Muslim, the Messenger, peace and blessings be upon him, would have asked about it at that point;