Chullin, Chapter 9, Mishnah 7. This chapter is devoted to the various forms of tum'ah that attach to an animal and to its parts, and our mishnah takes up the tum'at ochalin, the tum'ah of foods, in the case of a limb or a piece of flesh that is barely still attached to an animal. The mishnah traces what happens in three situations: while the animal is still alive, after it has been properly slaughtered, and after it has died on its own.
A limb hanging by a thread
The mishnah begins with the words "Ha'eiver vehabasar hameduldalin babeheimah": a limb, and a piece of flesh, left dangling from an animal. Imagine a leg cut through almost completely, and a chunk of meat torn nearly free, each still joined to the body by no more than a thread of sinew, while the animal itself goes on living. Concerning these the mishnah teaches "mitam'in tum'at ochalin bimkoman": they can take on the impurity of foodstuffs right where they hang, without any need to detach them first.
The force of the ruling emerges from the contrast. Suppose the limb had actually come free of the body. It would then be eiver min hachai, a limb severed from a living creature, and the torn flesh would fall under the parallel heading of basar min hachai. Eating either would be forbidden, and each would carry the impurity that such parts convey, no differently from a neveilah. In our case, though, separation never took place. Slender as the remaining attachment may be, both pieces still belong to a living animal, so nothing has yet entered the category of eiver min hachai, and neither its prohibition nor its form of tum'ah is in play.
On what grounds, then, are these pieces open to tum'at ochalin? Because at this point they have not lost their standing as food. A person could hand them over to a gentile, and the fact that a gentile is likewise barred from eating flesh torn off a live animal does not remove the substance from the category of edible matter. Whatever counts as food can absorb impurity from a tamei object that touches it, and can pass that impurity along to other food. That is the entire logic here: half-severed though they are, these portions remain ochalin, and since eiver min hachai has not yet taken hold, the impurity that governs them is the impurity of food.
A qualification follows: "Utzrichin hechsher", hechsher is required, the readying that first opens food to impurity. Like any other food, this limb and this flesh become susceptible only once moisture from the seven designated liquids has reached them.
After shechitah
"Nishchatah habeheimah", the animal was slaughtered. Now the dangling limb and flesh no longer need any separate act of preparation, for "huchsharu bedamah", they were made susceptible by the animal's own blood. At the moment of shechitah the blood pours from the cut and wets the carcass, and that moisture itself readies the meat to receive tum'ah. So teaches Rebbi Meir: "divrei Rebbi Meir".
Rebbi Shimon disagrees: "Rebbi Shimon omer lo huchsharu". In his view the blood that flows at shechitah does not accomplish hechsher, and the meat is not rendered susceptible to tum'ah by having been wet with it.
If the animal died on its own
The last scenario shifts the picture: "Meitah habeheimah", the animal expired on its own, from some cause other than shechitah, with the limb and the flesh still hanging from it. Of the flesh the mishnah rules "habasar tzarich hechsher", the dangling meat does need to be readied. No slaughtering blood flowed here to do that work, so moisture from the seven liquids must reach it before it can take on the impurity of foods.
The limb, by contrast, "mitamei", it transmits impurity, and the mishnah specifies the ground: "mishum eiver min hachai". Practically speaking, this limb had already been cut away, dangling as it was from the body. With the animal's death, the last shred of tissue that held it in place is dismissed as insignificant, and we regard the limb as having parted from the animal back while the animal was still alive. Hence the mishnah continues: "ve'eino mitamei mishum eiver neveilah", it is not treated as a limb belonging to a carcass. The animal lying there is indeed a neveilah, yet this hanging limb possesses a standing of its own, that of eiver min hachai, and it is not reckoned among the carcass's parts at all. Once more the mishnah signs off: "divrei Rebbi Meir".
Rebbi Shimon's final word
The mishnah closes: "veRebbi Shimon metaher", Rebbi Shimon declares it pure. He is not addressing this last case, for it would be difficult to speak of purity where a dead animal is involved. His comment reaches back to the opening case of the mishnah, the living animal with a limb and flesh dangling from it, which Rebbi Meir held to be subject to tum'at ochalin as they are. Rebbi Shimon disputes that ruling and frees them from the tum'ah of foods. His reasoning turns on the very point that had been used to establish them as food: a non-Jew is himself forbidden to eat such a thing, so it cannot be classified as food at all. Having no status as food, it has no tum'at ochalin, and Rebbi Shimon rules it pure.