What is the Mitzvah of "Anochi"?
Ramban and Seforno: That Hashem alone is G-d, the One who preceded the world and who created it, 1 and, since we accepted Him as our G-d without a medium, He has undertaken to be our G-d without a medium, and therefore we are obligated to Daven to Him alone and to serve Him directly - not through a medium. 2
Da'as Zekenim: We must know that He is the Creator and commander, and He is faithful to pay and punish.
What is the gist of "Anochi" as the first of the Aseres ha'Dibros?
Rashi #1: Refer to 20:2:3:1.
Rashi #2: They heard voices coming from all directions, from the Heaven 1 and from the earth. Lest you say that there are many Powers, Hashem said "Anochi", to make it clear to them that there is only One G-d.
Da'as Zekenim, Hadar Zekenim #1: Hashem appeared on the day of Matan Torah with myriads of angels, and Yisrael were astounded
Da'as Zekenim: Initially they heard Hashem from the east; they went to the east, and heard Him from the west; they went to the west, and heard Him from the north; they went to the north, and heard Him from the south; they went to the south, and heard Him from Shamayim
Why did Hashem say "Elokecha" and not "Elokeichem"?
Rashi: To give Moshe the opportunity, after Yisrael worshipped the Golden Calf, to claim that the prohibition against idolatry was confined to him alone. 1
Ramban: All the Aseres ha'Dibros were said in the singular, to implicate the individual who sins. 2
Hadar Zekenim, citing a Midrash, Moshav Zekenim #1: It is in the singular, for Hashem saw individuals who would transgress them
Which explains why Moshe stated below (32:11) in defense of Yisrael "Why Hashem, are you angry with your people?" You did not command them; you commanded me! (Rashi).
So that nobody should think that reward and punishment go after the majority, and if Yisrael are meritorious, he will be spared from punishment (Ramban).
Why did He link the Aseres ha'Dibros with Yetzi'as Mitzrayim?
Rashi: If only for the fact that Hashem took us out of Egypt, we would have been obligated to become subservient to Him.
Ramban: Yetzi'as Mitzrayim proves (a) Hashem's existence; (b) that He knows what is happening and supervises it; (c) that He created the world, 1 and (d) that He can do anything, which in turn, is proof (e) that He is Unique, 2 adding "Asher Hotzeisicha" to point out that Yisrael witnessed it firsthand.
Hadar Zekenim #1: I command you that I will be your Master, for I took you out of Egypt
Because if it had evolved naturally, nature could not possibly have changed its course (as happened in Egypt [Ramban]).
Without any intermediary - with reference to nature and to the celestial beings, as Yisrael declared after the drowning of the Egyptians (above, 15:27) "Zeh Keili v'Anveihu" (Seforno).
Lev Eliyahu (p.133): I.e. he thinks that we left naturally. Really, Hashem took us out to be the Am Segulah, i.e. miraculous Hashgachah Pratis which the nation, and with each person.
What are the connotations of "Anochi Hashem Elokecha"?
Rashi: At the Yam-Suf, Hashem appeared to Yisrael in the guise of a mighty Man of War; whereas here He appeared to them like a sage overflowing with mercy. So He began the Aseres ha'Dibros with "Anochi Hashem Elokecha" assuring them that He is One.
Ramban: It is a Mitzvas Asei, to believe that Hashem exists, who preceded time, and who created everything. He is our G-d and we are obligated to worship Him.
What are the connotations of "mi'Beis Avadim"?
Rashi #2 (in Tehilim): Hashem inserted this to stress the great favor that He had done them, that they should acknowledge that He is now their Master and they are obligated to accept Him as a G-d.
Seforno: In their capacity as slaves, they would sometimes have been unable to perform Hashem's Mitzvos; now that Hashem took them out of the house of slaves, they no longer had an excuse not to perform them.
Pesikta 3 (21, Aseres ha'Dibros): They were in a house (nation) of slaves, i.e. Mitzrayim, the son of Cham. All families from Cham are called slaves.
Rashi: Do not say that they were slaves of [the Egyptians, i.e. Pharaoh's] slaves