This song is recounting how everything we do on Shabbat is kadosh (think in terms of being transparent to its originating source) and is a sign (think that it is an indicating marker, like a road sign along the way). That is the repeating phrase (אות היא לעולמי עד בינו וביני) "It is an eternal sign between Him and me."
If you look at the Roshei Teivot of this repeating phrase, it emphasizes another association (אהל עבו היינו ע״ב ו) His thick tent (a reference to the clouds of glory, the thick clouds, which surrounded the Jewish people after leaving Egypt). His tent is also a reference to the Tent of Meeting, the Mishkan. It also has the implication that this was the place where the 72 (ע״ב) letter name was pronounced by Aharon HaKohen Gadol on Yom Kippur. And that all of this was to draw kindness (chesed "חסד" which is gematria of 72) into this world from above. This is accomplished through the middle path which is called "Tiferet" and is associated with Yaacov Avinu and the letter "Vav" which has a gematria of 6, like the sixth day. The conclusion and end of this middle path is the sefirah of Yesod which is what connects with and leads into Malchut "Kingship". Malchut being a reference to the Shabbat Kallah which we greet at the close of the sixth day in reciting Lecha Dodi and Eishet Chayil and also to our declaration the HaShem is King of the Universe in the recital of kiddush on Shabbat.
So we see an association between what was done coming out of Egypt, what was done in the Temple (the Mishkan is the Temple and the Temple is the Mishkan) and what we do every Shabbat.
The stanza in the song says, "To the first generation was given My holy one (meaning Shabbat). A wonder in giving a double portion of bread (the Manna) on the sixth. Thus it is with all sixths, you will double my sustenance."
This means that the concept of doubling the bread is related to six. In the Temple, that translated to the requirement that the Lechem HaPanim was 12 breads (2 times 6). One for each tribe that came out of Egypt. Part of the language for kiddish is to say 'in remembrance of going out of Egypt'.
Although the general custom of Ashkenazim is to use only two challot for kiddush, the practice of the Kabbalists, particularly those who follow the teachings of the Ari z"l, is to place 12 challot on the table for Shabbat kiddush. This is done in two arrangements of six, one on top of the other. This follows the precise wording from VaYikra 24:5-9. It says there that every Shabbat, without interruption, the children of Israel will arrange this bread before G-d as an eternal covenant. And it will be for Aharon and his children (meaning those who follow the practice and teaching of Aharon. Aharon is called Ish Chesed, and those who follow his teachings and practice are 'Chassidim'. Additionally, the entire Jewish people are called 'a nation of priests' like in Shemot 19:6.). And they will eat it in a holy place because it (the bread that is placed) is Holy of Holies to Him from the fire offerings of HaShem. And this is an eternal law (חק עולם).
The command from the Torah is that when you place the arrangement of bread, you will eat it. It is prohibited to fast on Shabbat because you must place the bread as part of the observance of Shabbat.
And this is following what happened to the children of Israel when they went out from Egypt. If they left the Manna over from Shabbat, it spoiled. They couldn't stockpile it for later usage.
The only exception to eating the bread on the day it was placed is Yom Kippur. If Yom Kippur falls on Shabbat, we are commanded to fast because we were specifically commanded to fast on that day. It is "Shabbat Shabbaton", the Sabbath of Sabbaths.
And that is precisely the text of song as it continues. "Marked as the law of G-d as a law (חוק כמ״ש ויקרא כד:ט) to His officers concerning it, 'to arrange the showbread before Him.' Therefore, to fast on it (the day you arrange it) is prohibited according to those who understand it (or understand Him), except for Yom Kippur when you will fast."
And this is precisely why this section from VaYikra is recited at the conclusion of Shabbat morning prayers in the siddur of the Alter Rebbe. It teaches the requirement to make the arrangement of twelve challah loaves for kiddush to fulfill the mitzvah properly.
This is also what is found in the Siddur of Rabbi Shabtai of Rashkov, the Siddur Kol Yaacov and the Siddur HaAri z"l from Zolkova.
