The idea you mentioned is found is in Ta'amei Haminhagim. There it says that the public Torah reading is a symbolic reenactment of the giving of the Torah at Sinai: The gabbai, the one who decides whom to call to the Torah, symbolizes Hashem, who chose the Jewish people to receive the Torah. The person called to the Torah represents the Jewish People. And the one who reads the Torah represents Moshe, the "broker" between the Jewish people and Hashem!
Sources:
- Ta'amei Haminhagim, Shabbat 328, quoting Levush 141:4
- Thanks to Rabbi Sholem Fishbane