The Hebrew word translated sign means "mark" or "evidence." The Sabbath day is the mark God gave His people to identify them as His own. By it, the folk of Israel would know the Source of their sanctification.
To sanctify is "to set apart for holy service," or more basically, "to make holy." God's purpose for Israel from the start was to set it apart from other peoples by giving it His laws and His statutes. God has a special relationship with Israel. Speaking through the prophet Amos to "the whole family [i.e., all the tribes] which I brought up from the land of Egypt" (Amos 3:1), God reminds the people that, "you only have I known of all the families of the earth" (verse 2). God revealed His law only to Israel. When He did so, He made it clear that Israel would "be a special treasure to Me above all people, . . . a holy [sanctified, set apart] nation" (Exodus 19:5-6), if the people "obey My voice and keep My covenant" (verse 5). The theme is repeated in Deuteronomy 7:6: "For you are a holy people to the LORD your God, . . . [who] has chosen you to be a people for Himself, a special treasure above all the peoples on the face of the earth." (See also Deuteronomy 14:2.)
God prefaces the "Holiness Code" of Leviticus 18 and 19 by commanding Israel to be separate from other nations. This meant acting in a way different from that of the Gentiles, not walking "in their ordinances." Leviticus 18:3-4:
According to the doings of the land of Egypt, where you dwelt, you shall not do; and according to the doings of the land of Canaan, where I am bringing you, you shall not do; nor shall you walk in their ordinances. You shall observe My judgments and keep My ordinances. . . .
In Leviticus 19:2, He makes His purpose clear: "You shall be holy [set apart], for I the LORD your God am holy." God's purpose, the intent behind all His laws, is to create a people like Himself (Genesis 1:26), a people sharing and reflecting His most salient attribute: holiness.
Sanctification is also the purpose behind God's often-denigrated physical laws. Consider, for example, the reason why God imposed the dietary law, as stated in Leviticus 11. God does not cite the maintenance of health as a reason to obey the dietary laws; the Scriptures do not specify that obedience of these laws will cause good health or prevent disease (though this is a secondary, albeit unmentioned, benefit). Rather, God concludes His dietary laws with a statement of His holiness and a command for His people to be like Him. Leviticus 11:44-45:
For I am the LORD your God. You shall therefore sanctify yourselves, and shall be holy; for I am holy. Neither shall you defile yourselves with any creeping thing that creeps on the earth. For I am the LORD who brings you up out of the land of Egypt, to be your God. You shall therefore be holy, for I am holy.
Obedience to God's law plays a crucial role in bringing about this sanctification. It is not that a people become sanctified (somehow, by God's grace) and, as a result, start obeying God's law. God's Word does not support the Protestant concept that sanctification imputed by God's grace mysteriously empowers one to obey His commandments. They have it backwards.
Rather, obedience to the law causes sanctification. Law-keeping and sanctification become intrinsically connected: To obey God's law is to be sanctified. By its nature, law-keeping brings about sanctification.
In a national context, God states that obeying His laws creates a people unlike others on the earth, a people set apart from others, a holy nation. National sanctification produces what Balaam saw in Israel: "A people dwelling alone, not reckoning itself among the nations" (Numbers 23:9).
If commandment-keeping separates people from the nations while connecting them to God, disobedience of God's law has exactly the opposite effect. Commandment-breaking separates a people from God, and connects them to the ways of the nations. Individuals who disobey God's law become like the "world," the kosmos of the New Testament (I John 2:15).