The Origin of Wrapping the Tallit
Question
Answer
Hello
Since there is an opinion that one does not fulfill the mitzvah of wearing a tallit unless it is 'wrapped', we perform during the blessing a form that is certainly 'wrapping' according to all opinions. Only afterward do we place it as a garment on the body, which fulfills the obligation of wearing according to most of the poskim who have ruled in halacha.
Source
Mishnah Berurah, Siman 8, S"K 4
During the wrapping, one covers his head with the tallit until it reaches his mouth and throws all four tzitzit to the left side for a distance of four cubits, and then removes the wrapping so that the tefillin is slightly visible, as mentioned above, and places each of the tzitzit in its place, as stated in paragraph 4, and this is the wrapping by which one fulfills the mitzvah according to all opinions.