The institution of sabbath can be led back up to the story of the Genesis (Gen 2:2-3), when God rested on the seventh day from all of his work, sanctified it. This cyclically returning order and frequency of time sequence was of such great significance for God that He declared it in the 10 Commandments:
Exod 20:8-11 "Remember the sabbath day, to keep it holy. Six days shalt thou labour, and do all thy work: But the seventh day [is] the sabbath of the LORD thy God: [in it] thou shalt not do any work, thou, nor thy son, nor thy daughter, thy manservant, nor thy maidservant, nor thy cattle, nor thy stranger that [is] within thy gates: For [in] six days the LORD made heaven and earth, the sea, and all that in them [is], and rested the seventh day: wherefore the LORD blessed the sabbath day, and hallowed it. "
(see also Deut 5:12-14)
In Deut 5:15 and 15:15 we can find a reference to the fact that the redeem from the Egyptian bondage was a kind of sabbath day for the people of Israel, implicating the spiritual importance of sabbath, the rest. According to this word, Israel had to remember of the exodus from Egypt with the sabbath.
Sabbath is mentioned, recorded in the first place among the feasts in Lev 23:3. The tenth day of the seventh month, the day of atonement was regarded as the sabbath of sabbath (sabbath of rest) in the Old Testament (Lev 16:31, 23:32 - Yom Kippur).
Furthermore, there is still a remarkable phenomenon in the Old Testament when in every 7th year it became possible for the Hebrew servants serving in Israel to be freed, go out free (Exod 21:2-6; Deut 15:12-18) as well as that order of God, which commanded the sabbath of rest onto the land when nothing was grown or produced on it (Lev 25:4). The jubile was the 50th year after the expiration of the 7 x 7 years (7 sabbath-years), during which freedom was given to everyone in Israel, so everyone could return to its own family and gained his possession back. (1 sabbath-year = 7 years) (Lev 25:8-13)
A kind of release is formulated in the spiritual importance of sabbath, which is summarized by prophet Isaiah as follows:
"This is the rest wherewith ye may cause the weary to rest; and this is the refreshing." (Isa 28:12)
Prophet Isaiah speaks about the real content of the fast in part 58, which is linked to the message of sabbath in my opinion:
"Is not this the fast that I have chosen? to loose the bands of wickedness, to undo the heavy burdens, and to let the oppressed go free, and that ye break every yoke? Is it not to deal thy bread to the hungry, and that thou bring the poor that are cast out to thy house? when thou seest the naked, that thou cover him; and that thou hide not thyself from thine own flesh? "(Isa 58:6-7)
Still in 58:13-14, the following is revealed as a part of the fast:
"If thou turn away thy foot from the sabbath, from doing thy pleasure on my holy day; and call the sabbath a delight, the holy of the LORD, honourable; and shalt honour him, not doing thine own ways, nor finding thine own pleasure, nor speaking thine own words: Then shalt thou delight thyself in the LORD; ...” (Isa 58:13-14)
If we interpret that in this word the pleasure is the sin (the sinful human nature) or the act of sinning and sabbath means staying away from sin or it is the conversion (repentance) itself, then we can regard that not to act sin, to give it up or to stay away from sinful human nature, to crucify it by acting good is what a really kind sacrifice to God.
So that truth of the New Testament is included in the Old Testament already, which suggests the spiritual message of sabbath.
The Gospel of Matthew, Mark and Luke also recall when Jesus explains the true message and significance of sabbath in relation to an event when the disciples plucked the ears of corn on a sabbath day, even referring back to David who was in need and ate the shewbread in order to survive (Matthew 12:1-8, Mark 2:23-28) as well as with reference to a case of healing on a sabbath day focusing on the respect of human life against the importance of sabbath (Luke 13:11-16). Jesus expresses that sabbath is not more important than man, but it is for him/her.
To sum up, sabbath is rest, calm, comfort, release, forgiveness, mercy, conversion, what is more according to Heb 4:9-11 it is salvation itself or the saved, blessed state:
"There remaineth therefore a rest to the people of God. For he that is entered into his rest, he also hath ceased from his own works, as God [did] from his. Let us labour therefore to enter into that rest....".