The commandment to wear the tzitzit of the tallit is one that comes directly from Biblical sources. The earliest Biblical mention of a tallit comes from the period after the Exodus when Moses and the Hebrew spent 40 years wandering the desert.
The reference specifically comes from the Book of Numbers. The specific verses where the tallit is first mentioned are among those that are included in the Shema. It is here where G-d commands Moses to instruct the people to wear the fringes. The verses referenced are of course Numbers, 15:38-40.
38: Speak unto the children of Israel, and bid them that they make them fringes in the borders of their garments throughout their generations, and that they put upon the fringe of the borders a ribband of blue:
39: And it shall be unto you for a fringe, that ye may look upon it, and remember all the commandments of the LORD, and do them; and that ye seek not after your own heart and your own eyes, after which ye use to go a whoring:
40: That ye may remember, and do all my commandments, and be holy unto your God.
The commandment to don the four corners of the tzitzit is repeated once more in the Book of Deuteronomy.
Interesting enough, the entire purpose of wearing the tzitzit of the tallit is to remember the 613 Mitzvot or Commandments that G-d gave to Moses for the Jews to follow. It is largely for this reason that it is a custom to don the tallit when being called up to the Torah, which of course contains the 613 Mitzvot from G-d.
The significance of the Tzitzit’s connection to the 613 commandments of the Torah is one that has been studied extensively. Many Rabbinic sources applied the practice of gematria, which assigns a numerical value to each Hebrew letter, to the meaning of the tallit. They found through the usage of gematria that the value of the word Tzitzit is 600. They then surmised that it was symbolic if added to the 5 knots and 8 strings of each fringe they got the number 13, which they then added to 600.