I have heard several reasons for braiding Challot for Shabbat. The three braids are symbolic of the commands to observe Shabbat that appear in the Ten Commandnments One braid represents the word "Zachor" - "Remember." A second braid represents the word "Shamor" - "Guard." The third braid is for "b'Dibbur Echad" - that these commands of "Remember" and "Guard" were said by God simultaneously and as one unit.
Another reason is that Shabbat signifies and reminds us of three different concepts: The Creation of the World, the Exodus from Egypt and the Messianic Era. This is also the reason for three distinct separate Amidot - Silent Prayers - on Shabbat, as opposed to the weekday Amidah which is of identical wording three times a day (the theme of the fourth prayer of Shabbat - Mussaf ("additional") is said for the additional Temple sacrifice for Shabbat, and also applies on Festivals.) This idea also provides an understanding for the three meals eaten on Shabbat.
By the way, the "Challah" mentioned in the Torah is not referring to the Challah that we eat on Shabbat and Yom Tov. It refers to the command to separate a small amount of the dough that one kneads when baking bread. In Temple times this portion of dough (called "Challah" by the Torah) was given to the kohanim, the priestly tribe, who were responsible for the Temple service. Today there is a rabbinical command to separate "Challah" from the dough and burn it, since in order to eat it there is a requirement for the kohanim and the Challah to be ritually pure - a state that does not presently exist.
