Good questions. Braiding Challah for Shabbat and Yom Tov is a widespread custom but is not an obligation. The simplest answer to why we braid the Challa is that is looks fancier! This is known as “hiddur mitzvah” – to do a mitzvah in a beautiful way.
The number of strands may vary according to one’s personal “taste” and family custom. I personally have seen 3, 4 and 6 strands in a Challah.
Most Challahs have three strands in my experience. Why three? My wife says it’s because this is the simplest and most practical way to make a braid, but I have also heard several reasons for braiding the Challah with three strands for Shabbat:
The three strands are symbolic of the commands to observe Shabbat that appear in the Ten Commandments. One strand represents the word "Zachor" - "Remember." A second strand represents the word "Shamor" - "Keep." The third strand 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.