The custom was once that the talis was the main garment a man wore. While that is obviously not the case anymore, we do wear a talis katan (small talis) to keep the mitzva of tzitzis with us the whole day. (It is a shmira - reminder or spiritual protection against forbidden relations.)
However, it is highly questionable if a regular talis katan1 fulfils the Torah's requirement of a real garment - one that can cover the body properly - and it therefore became practice to wrap (ituf) one-self in a proper talis (called talis gadol - large talis) once a day. The time chosen for this was set to be that of shacharis, which is the most elaborate and foundational prayer of the day.
Our days' talis therefore has a special elevated status, being both the full-scale performance of a Torah mitzva, and also a garment set aside for the holiest time, i.e. prayer. (That is why we do not wear it for Tisha B'Av shacharis - as a sign of mourning.) The talis is used as a sign of higher status: To carry a Sefer Torah. By the one leading mincha. Yom Kippur by night, and the whole day too. To enhance this mitzva, many decorate their taleisim elaborately with silver plates or woven bands.
When wearing a talis, it is customary to hold on to the talis even beyond shacharis: On Yom Tov, mincha is sometimes prayed right after shacharis (before the meal) in which case many keep the talis on for mincha too. Some keep it on until after Shabbos day kiddush. If shacharis is followed by a bris mila, many will keep their tallis on for this too.
Finally, some may find that their concentration is enhanced by using the talis to shield off the world during prayer.
1. A very large and long talis katan in accordance with the Gr"a may.