I often hear that מוצאי שבת means "departure of the Sabbath," but it seems to me that the phrase for this should be צאת שבת (or יציאת שבת).
The phrase מוצאי שבת seems to me to mean המוצאים של שבת, which would literally be "the things of Shabbos that proceed forth." (I'm thinking of מוצא as in על כל מוצא פי ה' יחיה האדם.)
Am I understanding this correctly or am I missing something? If so, then what are the "things that proceed forth" and why is Saturday night called this?
This may be too much for an unsourced "drush" but here goes:
The אור החיים on the words כי ששת ימים עשה ה' את השמים ואת הארץ points out that the verse does not say בששת ימים but ששת ימים - not "in six days He created" rather "six days He created." He explains that when Hashem created the world, He gave it the "juice" for only those first six days, and Hashem continues to create the world for 6 days at a time, every Shabbos. Shabbos is the source of the creation, the השפעה, of the following week. Therefore we celebrate the events of the upcoming week on Shabbos (Shabbos Mevarchim, etc).
Therefore, every week is the תוצאה of the preceding Shabbos, and could be described as "things of Shabbos that proceed forth" - after Shabbos is the result of Shabbos.
In the Selichos of the first day of Selichos before Rosh Hashana we find במוצאי מנוחה קדמנוך תחלה, which is translated (Artscroll) "As [the day of] rest departs, we come before You first of all".
Also, one of the Zemiros which is sung after Shabbos starts with the words במוצאי יום מנוחה which the Artscroll Siddur translates as "At the departure of the day of contentment".
Thus מוצאי שבת would be translated as "the departing of Shabbos".
Jastrow's dictionary includes מוצָאי as the plural construction of מוצָא (exit), and presents יצא as the equivalent in biblical Hebrew... though some searching shows that מוצָאי is sometimes used in biblical Hebrew as well (Psalms 65:9, Ezekiel 12:4, Ezekiel 44:5).
The phrase מוצאי שבת is utilized widely in the Mishnah and Talmud, and rabbinic literature generally (for example the Mishnah Berakhoth 2:5 comes immediately to mind).
Like yEz's answer, this answer is not sourced; it is simply a suggestion.
מוצאי שבת refers to Saturday night. In my experience, there is a certain feeling on Saturday night that is different from the rest of the week. It's not Shabbat anymore, but there is a bit of the feeling of Shabbat left over. You're probably still wearing your Shabbat clothes and you have a melaveh malkah. Maybe that Shabbat-like feeling is what goes out of Shabbat?
If I had to guess, I would say the original term was מוציא שבת which translates to the "one which takes out Shabbos" which is figuratively what the night following Shabbos does. Then somehow, the א and י switched places.
Somewhat of a support to this theory is the way I've heard some Lubavitchers (may be true for others from the Russian region as well) pronounce the term: the say Motzi Shabbos which is actually the pronunciation for מוציא rather than מוצאי.
As far as I know, the phrase that is often mispronounced as “motzi shabbos” is actually “motza’ei shabbos”— מוצאי שבת, which is the סמיכות form of המוצאים של שבת, “the departures of shabbos” in plural. Moreover, strictly speaking, it only refers to the departure of שבת, which makes a lot of sense in contexts like ״סדר למוצאי שבת״, but not Saturday night (for lack of a better unambiguous phrase). Therefore, the phrase should be ״אחרי השבת״ or something similar, but few Ashkenazim are well-versed in דקדוק, so the misconstrued “מוצאי שבת” stuck.
