The matzot are symbolic of the three castes of Jews: Priests, Levites and Israelites; all of whom were liberated from Egyptian bondage.
They also commemorate the three measures of fine flour that Abraham told Sarah to bake into matzah when they were visited by the three angels.1According to tradition, the angels’ visit was on Passover.
They also represent our Patriarchs, Abraham, Isaac, and Jacob, in whose merit we were redeemed from Egypt.
On a practical level, three matzot are needed so that when we break the middle matzah, we are still left with two whole ones to pronounce the Hamotzie blessing (as required on Shabbat and Holidays. See The Two Loaves).
For the mystical spin on the matzah numerology, see The Matzot and Four Cups of Wine.