Why does the Torah insert Shabbos in the Parshah of Mo'adim?
Rashi: To teach us that, on the one hand if someone desecrates Yom-Tov, it is as if he desecrated Shabbos, whilst on the other, if he observes Yom-Tov, it is as if he observed Shabbos.
Ramban #1: Because Shabbos also falls under the category of 'Mo'ed' and can be termed "Mikra Kodesh". 1
Ramban #2: "Mo'adei Hashem" (in Pasuk 2) refers specifically to the Yamim-Tovim listed later. And the Torah is coming to draw a distinction between the Mo'adim of Beis-Din and Shabbos, 2 inasmuch as only Meleches Avodah is forbidden on the former, all Melachah is forbidden on the latter. 3
Ramban #3: It is coming to teach us that if Yom-Tov falls on Shabbos, it does not override it, to permit cooking - Shabbos remains intact. 4
Divrei Eliyahu: Here, 'Shabbos' refers to Yom Kipur, when all Melachah is forbidden. The 'six days' are the other Yamim Tovim, on which Melachah of Ochel Nefesh is permitted.
Ramban: The Torah goes on to separate the Yamim-Tovim from Shabbos, by inserting the words "asher Tikr'u osam Mikra'ei Kodesh", which it did not insert in Pasuk 2.
Ramban: In Pasuk 4, the Torah repeats "Eleh Mo'adei Hashem ... " because it interrupted with Shabbos.
Ramban: Proof of this and the following explanation lies in the fact that the Pasuk does not mention Korban by Shabbos, and then at the end of the Parshah (in Pesukim 37, 38) it concludes "Eileh Mo'adei Hashem ... Mikra'ei Kodesh Lehakriv Isheh la'Hashem ... Mil'vad Shabsos Hashem" - A clear proof that Shabbos is not included in the Mo'adim. See Ramban, who brings a second proof.
Seforno: Pesachim, 117b.
Seforno: As the Gemara states in Rosh Hashanah, 25a - 'You (Beis-Din, your declaration of Rosh Chodesh is valid) even b'Shogeg! You, even b'Mezid! You, even if you were tricked!'
Why does the Pasuk conclude "b'Chol Moshvoseichem"?
Ramban (15, citing the Mechilta): To preclude preclude Hashem's dwelling (the Beis-ha'Mikdash) from the prohibition of making a fire. 1
Ramban (15, citing Kidushin, 37b): To teach that, even though the Torah inserts Shabbos in the Parshah of Mo'ados, it does not require the Beis-Din to sanctify it. 2
Seforno: Even though the length of a day differs from place to place, 3 we must keep Shabbos from nightfall to nightfall according to the way Hashem who created light and darkness fixes it in that location.
Why does it say "Te'aseh Melachah" (passive)?
Moshav Zekenim (Shemos 35:2): When Yisrael merits, our work is done through others.