You shall offer salt on all your offerings. (Leviticus 2:13)
The bringing of offerings to G-d in the Temple was part of the process of repentance and atonement for sinful behaviour.
Rabbi David Kimchi (known as Radak and d. 1235) explains that the altar (mizbeach) on which sacrifices were offered in the Temple can be described as the Table of G-d (Radak to Malachi 1:7). Since we no longer have the Temple and therefore no longer have the opportunity to bring offerings to G-d, the tables on which we eat and the meals we enjoy act as substitutes for the altar and the offerings respectively (Brachot 55a).
The Shulchan Aruch (Code of Jewish Law, written in 1563) writes that one should add salt or relish to the bread before eating it. The inclusion of relish (which is understood to be any sort of dip eaten with bread) indicates that this has less to do with the altar in the Temple and more to do with the requirement that the bread must have a pleasant taste in order for the blessing made before eating it to be valid. The Shulchan Aruch continues to say that if the bread is made from pure flour or already seasoned with salt or spices, no salt or relish is required.
Nevertheless, Rabbi Moshe Isserlis (known as the Rema d. 1572), based on the above verse explains that salt should be placed on the table at the beginning of the meal, before breaking bread as the salt reminds us that our table is like the altar.
This may explain why it has become customary to use salt on Shabbat even though many do not during the week. Although there appears to be no obligation to do so since our challot already contain salt, the salt serves to remind us that eating a meal is more than a means to eat food. Just as the altar taught us to dedicate ourselves to G-d, through the company of friends and family the essence of Shabbat takes the most physical act such as eating and turns it into a most beautifully spiritual experience.