In this Mishnah, we'll talk about a case where an avaryan, a sinner, is put to death not by the Beit Din (court), but by the people around him. The Mishnah says: Somebody who steals a sacred Keli, a utensil or vessel, from the Beit HaMikdash. If somebody curses the name of Hashem, then, using an Avodah Zarah (idolatry)—for example, he says, “My Avodah Zarah,” and names it Avodah Zarah—he curses Hashem. And then example number three: Somebody who has inappropriate relations with an Arami (Aramean) woman, a woman who's an idol worshiper, and has relations with her in public, in front of ten men of Yisrael.
Then people who are very pious and worried about the honor of Hashem are allowed to attack and kill all of these people. These people are undermining the sanctity, the Kedushah, the holiness of the Jewish people. So if somebody sees them doing it, they're allowed to attack them and kill them. If a Jewish man is having relations with an idol worshiper, he's supposed to receive Karet (spiritual excision) from Shamayim (heaven), and he's also Chayav Malkot (lashes)—he's going to get lashed. But that's only if he's with her privately. Here, when he's doing it publicly, he's committing a very, very egregious, very high-level sin, and therefore the zealot, somebody who cares about the honor of Hashem, is allowed to kill him. But if the sinner is defending himself and kills that Kana'i (zealot), then we do not execute him for murder, because we view him as protecting himself. So these are the three cases: if a person sees that aveira (sin), he could go attack and stop it in order to restore the honor of Hashem.
A Kohen (priest) was serving in the Beit HaMikdash when he was Tamei (ritually impure), and he was doing this on purpose. His brothers, the other Kohanim, don't bring him to Beit Din in order for him to receive the Malkot that he's supposed to get. Rather, the young Kohanim were just starting to grow beards. They remove him outside the courtyard, outside the Azara, the courtyard. They don't want to have any Tum'ah (ritual impurity) in the courtyard of the Beit HaMikdash. And they crack open his skull, his head, with wooden logs that they have over there in the Beit HaMikdash that they would use as firewood on the Mizbeach (altar). They don't want to wait for weapons or something else to be brought. They would kill this person right away for bringing something negative to the Beit HaMikdash of Tum'ah. And again, we're talking about somebody who's trying to disgrace Hashem by doing that on purpose.
“Zar she-shimesh baMikdash”: if a non-Kohen serves in the Beit HaMikdash, Rebbe Akiva says that the Beit Din executes him by strangling him. Chachamim (Sages) omrim bidei Shamayim. The Chachamim disagree and say that he's put to death by Hashem. The Beit Din does not go kill him, and other people don't kill him either; he's put to death by Hashem.