an animal killed and ripped open by a wild beast, and so the mangled remains it leaves behind
This noun is built on the verb "to tear" (טָרַף) and names one very specific thing: an animal a predator has brought down and torn open. English has no noun for that, so translators reach for a phrase, and the phrase changes from version to version: "that which was torn", "torn by beasts", "torn to pieces", "torn flesh".
Most of the nine places the word appears are food laws, where it stands beside a second word for an animal that simply died on its own (נְבֵלָה). The pairing is deliberate: between them the two words cover every way an animal could end up dead without being properly slaughtered. Meat like this had never been drained of its blood, and a wild animal had already fed on it, so anyone who ate it was unclean until evening. Priests were held to the rule strictly, and Ezekiel offers his lifelong avoidance of both kinds of meat as proof of his obedience.
Two other uses have nothing to do with diet and everything to do with money. When Jacob lists what he endured in Laban's service, and when Israel's case law settles a quarrel over a borrowed animal, the torn carcass is the physical evidence in a dispute about who bears the loss. A herdsman who could produce the mangled remains had shown that the animal was taken by a predator rather than stolen or sold.
One use is pure poetry. Nahum pictures Nineveh as a lion's den heaped with torn flesh, and sets this word alongside the ordinary word for what a hunting animal catches (טֶרֶף). The image is exactly what the legal texts are worried about, only seen from the lion's side of it.