“To know”: the purpose of the Sabbath is the True Knowledge: “to know.”
“For it is a sign between Me and you”:
A sign is intended to signify, which means to teach some knowledge. This, therefore, requires that Israel should utilize the Shabbos (Sabbath, Shabbat, ed.) for considering these two fundamental principles:
1) the creation of the Universe by the Creator,
2) and the election of Israel as HaShem’s holy people (“I HaShem make you holy”).
But this second principle is in itself composed of two very important parts.
A) Israel must always seek to remind itself that it is a holy nation, entirely unlike all other nations. The entire purpose of Israel is different, and the entire lifestyle of a Jew must always be different. The Sabbath is one of the most conspicuous ways of making us aware of our function as a people of “holy men” (22:30).
B) The Israelite must forever be mindful of the endless debt of gratitude to Hashem for electing Israel to become His holy people. He makes us holy by giving us His laws (“And you shall remember all My commands and you shall do them and you shall become holy to Hashem your G-d”—Bamidbar 15:40).
Thus we learn that the function of the Sabbath is to make us aware of the election of Israel by HaShem as His holy nation.
But - we read soon after (31:17) “For it (the Sabbath) is a sign forever that in six days Hashem made the heavens and the earth.” If the Sabbath signifies these two principles, then we understand that they are equal to each other.
Thus, Israel is made holy in order to proclaim that Hashem made the Universe from nothing, and the Universe was made in order that Israel should become the holy nation which would elevate HaShem as the Creator of the Universe.
This function “to know that I HaShem make you holy” is so very important, that it explains the sequence of the Torah-laws.
The first laws were the Pesach-laws, given in Egypt; for the Pesach symbolizes the choice of Israel as Hashem’s first-born son who is chosen forever to serve Hashem as His holy people. The law of Shabbos followed soon after before they came to Sinai, to demonstrate the election of Israel forever (“Between Me and the sons of Israel it is a sign forever”—3l:l7).
Pesach and Sabbath preceded Sinai, because the choice of Israel by HaShem is of paramount importance in the Torah ideology. (A Nation is Born)