What is the connection between this Pasuk and the previous one?
Ramban (on Pasuk 1): None! It is the commencement of Moshe's direct speech to Yisrael, and the word 'Vayomer' is missing from the beginning of the Pasuk. 1
What are the implications of "Rav lachem Sheves ba'Har ha'Zeh"?
Rashi #1: It implies that Yisrael had spent enough time at Har Sinai (and it was time to move on).
Rashi #2: 'You have earned much reward during your stay at this mountain! 1 - You built the Mishkan, the Menorah and the other Keilim; you received the Torah and you appointed the Sanhedrin, officers over thousands, officers over hundreds! (And it will be detrimental for you to remain there any longer' - Targum Yonasan).
See Sifsei Chachamim.
What exactly is "Chorev"?
Ramban #1: The name of the mountain is Har Sinai. 1 Chorev is either the place or town next to it where Yisrael stayed, 2 or the entire area was called Chorev, in the middle of which Har Sinai was situated. 3
Ramban #2 (citing a Midrash): Chorev is the location where the Mishkan was built. 4
Ramban #3 (citing the Ibn Ezra): Chorev is synonymous with 'Har Sinai'. 5
Ramban: And Midbar Sinai is called after it.
Ramban: As that is the place which the Torah is referring to when it writes in Sh'mos 19:2 "Vayichan Sham Yisrael Neged ha'Har". When it wrote there (3:1) "Vayavo el Har ha'Elokim Choreivah", it means that he (Moshe) came to Chorev which was next to Har Sinai, which explains why the Pasuk continues "Asurah Na ve'Er'eh" meaning that he would go from Chorev to Sinai to see the burning bush.
Ramban: Which will explain the Pasuk in Mal'achi, 3:22, which refers to the Torah that Hashem commanded Moshe in Chorev, the Chukim and the Mishpatim - incorporating all that Hashem had commanded him on Har Sinai and in the Ohel Mo'ed, both of which were in Chorev.
See Ramban, DH 'v'Ra'isi b'Yelamdeinu ... '.
Ramban: And it is called 'Chorev' because, the bushes (which is why it is called Sinai - from the word 'S'neh') grow on it as a result of its dryness ('Chorev'). The Ba'al ha'Turim too, citing a Midrash, assumes that "Chorev" is synonymous with Har Sinai, since he attributes the missing 'Vav' in "Chorev" to a hint that, if Yisrael do not keep the Torah (which was given on it), Hashem will bring a sword (Cherev) against them.