Shabbat reveals the special connection between God and Israel, as is stated:
Nevertheless, you must keep My Shabbatot, for it is a sign between Me and you throughout the ages, that you may know that I the Lord have consecrated you. You shall keep Shabbat, for it is holy for you…. The Israelite people shall keep Shabbat, observing Shabbat throughout the ages as a covenant for all time; it shall be a sign for all time between Me and the people of Israel that in six days the Lord made heaven and earth, and on the seventh day He ceased from work and was refreshed. (Shemot 31:13-17)
The Sages report that “God said to Moshe: ‘I have a wonderful gift in my treasury, and its name is Shabbat. I wish to give it to Israel. Go and inform them”’ (Beitza 16a). They further state: “All the mitzvot that God gave to Israel were given publicly, except for Shabbat, which was given in private, as it says: ‘it shall be a sign for all time between Me and the people of Israel’” (ibid.). They then raise a question: Isn’t the mitzva of Shabbat found in the Ten Commandments, which were given very publicly? They respond that the true, inner meaning of Shabbat, which reveals the divine nature of the world, is something that cannot be revealed publicly. It is something special that is linked to the special connection between God and the Jews. In their words: “God gives people a neshama yeteira [lit. “expanded soul”] before Shabbat, and after Shabbat he withdraws it from them” (ibid.). By means of this neshama yeteira, the Jews are able to grasp the godly significance of the world and their special role within it.
This explains the Sages’ statement: “A non-Jew who observes Shabbat is deserving of the death penalty” (San. 58b). They further state:
In human society, if a king and a noblewoman are sitting and talking with each other, wouldn’t one who interrupts them be deserving of death? Similarly, Shabbat is something shared by the Jews and God, as [Scripture] states: “between Me and the people of Israel.” Therefore a non-Jew who insinuates himself between them before undertaking circumcision deserves death. (Devarim Rabba 1:21; see Harĥavot 25:1)
In order to express the Jews’ great love for Shabbat, which is like that of a royal bride at the side of her groom, there is a Jewish custom to go outside before sundown on Friday to greet Shabbat, just as one goes out to greet an honored guest. The Gemara tells us that R. Ĥanina would put on his best clothes and greet Shabbat with the declaration: “Let us go and greet the Shabbat queen.” R. Yannai would wear his fancy clothes and greet Shabbat by declaring: “Welcome, O bride; welcome, O bride.” This inspired R. Shlomo Alkabetz’s magnificent poem Lekha Dodi: “Go, my beloved (Israel) to greet the bride; let us greet Shabbat.”