We always dip the bread in salt, not only on Shabbat.
Our table is considered an altar (see Ezekiel 41:22 and Ethics of our Fathers 3:3), and in the Holy Temple salt was offered together with every sacrifice (Leviticus 2:13).
Salt never spoils or decays, therefore, it is symbolic of our eternal covenant with G-d. That's why the verse refers to it as "the salt of your G-d's covenant."
[Salt also adds taste to everything. Our bond with G-d is supposed to add meaning and flavor to every moment of our lives; even when we are not directly involved in spiritual pursuits.]