After "leining" the Torah, we pick up the Torah scroll to show everyone. Why do we do this?
It's actually not necessarily after the leining; the Sefardic custom is to do this beforehand, when the Torah is taken out of the Ark. (Indeed, for this reason the Shulchan Aruch - whose author, R' Yosef Caro, was a Sefardi - puts the laws of hagbahah in sec. 134 of Orach Chaim, preceding the laws of reading the Torah in secs. 135ff.)
Either way, though, the Shulchan Aruch (134:2), quoting Maseches Sofrim 14:14, says that the reason is so that everyone can bow respectfully towards the Torah and say "Vezos Hatorah" ("and this is the Torah..." - "this" implying something that you actually see).
Magen Avraham (134:3) adds, citing "Kavanos" (Kabbalistic writings from the Arizal's school), that when one sees the actual words of the Torah, this draws down a great Divine light upon him or her.