Why does it say "Amarti Ani b'Libi Al Divras Bnei ha'Adam"?
Rav Sadya Gaon: I said in my heart, according to what people say, since [also] Hashem revealed and publicized [that they are animals], and so they themselves will see that they are animals.
Rashi: I saw from people's speech that they have pride, to conduct with authority over those lower than themselves.
Ibn Ezra: "Al Divras" is due to, like "Al Divrasi Malki Tzedek" (Tehilim 110:4) - because you are Malki Tzedek.
Rashbam: It is regarding people who are sinners.
Ri Kara: Al Davar (due to) a small sin that they did.
Seforno: I know that perforce Hashem will judge also Yisrael, for He said "Na'aseh Adam b'Tzalmenu ki'Dmuseinu" (Bereishis 1:26), in order to choose for them the Divine part - "Ani Amarti Elokim Atem" (Tehilim 82:6).
Metzudas David: It is about men's matters 1 - how to conduct and what to do.
Metzudas Tziyon: "Al Divras" is about the matter, like "Al Devar ha'Kesef" (Bereishis 43:18). Here it says Divras, for it is Samuch to Bnei ha'Adam, like "Avi v'Zimras Kah" (Shemos 15:2).
What is the meaning of "Levaram ha'Elokim"?
Rashi: Hashem clarifies 1 that their authority is nothing.
Ibn Ezra: The prefix Lamed in l'Varam is [for, like "Imri Li Achi Hu" (Bereishis 20:13). The grammatical form of Varam is like "Ka'asher Ra'am 2 " (Bereishis 32:1). It is like "Beru Lachem Ish" (Shmuel I, 17:8). Even though a verb in past tense does not have the prefix Lamed anywhere else, Lamed is like Beis, and we find like this, just once, "ba'Hechin Lo David" (Divrei ha'Yamim II, 1:4).
Rashbam: He will clear them from the world. Levaram is like "u'Varosi Mikem ha'Moredim veha'Posh'im Bi" (Yechezkel 20:38). It is Pi'el conjugation, from the root Barah 3 , just like Lechalam is from "Lechalah" (ibid. 13:13). The Reish should have a Dagesh, just Reish does not receive a Dagesh.
Ri Kara: Hashem decided to choose them - He will destroy them and [others] will dwell in their place.
Rid: Hashem arranged His world that man has no advantage over animals, so people will know to clarify themselves and see that they are animals, like verse 10 explains.
Metzudas David: They will clarify and discern the grandeur of Elokim. Metzudas Tziyon - Levaram is an expression of Birur (clarifying) and discernment.
Ibn Ezra: I.e. it is as if it says Levarer Osam, Pi'el conjugation; the middle letter of the root should have a Dagesh (just Reish does not receive a Dagesh). This is difficult, for in Pi'el, the middle letter of the root is not swallowed (omitted)!
I.e. it is like Barah Osam, just like there it is like Ra'ah Osam. (PF)
Rashbam: You cannot say that the root is Bar. If so, it should have said Lehaviram (Hif'il), just like from the root Shav it says Lehashivam.
What is the meaning of "[v'Lir'os] she'Hem Behemah Hemah Lahem"?
Rashi: Even the officers and kings are like other animals to themselves.
Ibn Ezra: Even though Hashem chose them over all living beings, they consider themselves like animals. [The prefix Kaf is omitted,] like "v'Ayir Pere Adam Yivaled" (Iyov 11:12), "Esh Ocheles Hu" (Devarim 4:24).
Rashbam: Hashem causes them to view themselves like animals.
Ri Kara #1: They will see that even in their deeds 1 , they are like animals. People are astounded when they see "Rasha Machtir Es ha'Tzadik" (Chabakuk 1:4; he surrounds him to harm him] Yirmeyah said "Tzadik Atah Hashem Ki Ariv Elecha "Tzadik Atah Hashem Ki Ariv Elecha Madu'a Derech Resha'im Tzalechah" (12:1). Hashem showed that they complain groundlessly.
Ri Kara #2: Hashem decided that these will leave and others will come and dwell in place of them, even though they are bigger Resha'im, so they will know that they are like animals, even in their deeds.
Seforno: They themselves will see that if not for their intellectual Tzelem Elokim, they are mere animals.
Metzudas David: They will see intellectually that they themselves are like animals - but compared to Elokim, they are nothing.
Magihah (in Toras Chayim): I.e. what happens to them.