Aruch Hashulchan (Orach Chaim 285:2) suggests that it is in commemoration of the three times that the generalities and the specifics (כללות ופרטות) of the Torah were repeated (Sotah 37b): once at Sinai, once by Hashem speaking to Moshe from the Tent of Meeting, and a third time by Moshe in the plains of Moav. The reason for doing the Targum one time, he says, is that it too was given at Sinai, and is necessary (or was, in Talmudic times) for the person to understand what he's learning.
The commentary Magen HaElef on Siddur R' Amram Gaon offers a different reason, and also uses it to explain why the reward for doing this is long life. It goes back to the time when people who got an aliyah were expected to read their own portion (as is still done in Yemenite communities); to do so takes prior preparation - reviewing the portion at least twice. If one is careful to do שמו"ת, then, he will always be able to receive an aliyah if one is offered to him, and will thereby merit the Torah's blessing of long life (Deut. 30:20).