Why do we wear a tallit at kol nidre?
And if one arrives late to kol nidre and night has fallen already, should he put one on?
Summarizing from p. 349 of this book:
The custom originated from Rav Meir of Rothenberg who stated that on the night of Yom Kippur we recite the 13 "middot" - Attrubutes of Mercy, multiple times during our tefillah (after the Amidah). There is a midrash (mentioned in Talmud Rosh Hashannah 17b) that G-d wrapped himself in a tallit, similar to a Chazzan when he taught Moses these attributes. Therefore, we imitate what G-d did when we recite them.
My curiosity - we can apply this "argument" to the recital of selichot, esp. the 1st one which is said at night. There, too, we recite the 13 middot multiple times, yet we do not wear a tallit, then. Perhaps, Yom Kippur is special?
Nitei Gavriel Hilchos Yom Hakipurim page 181 note 24 in the name of Siach Yitzchak that we wear a Talis at Kol Nidrei based on the Yerushalmi Chagiga 1:5 that says that one needs to wear a Talis at the time of Hataras Nedarim.
He goes on to say that one should only make a Bracha on the Talis up to the time he would normally Daven Mincha.