The cups parallel the four expressions in the Torah which describe our freedom from Egypt. The first cup, which also serves as Kiddush, parallels "I will take you out," when God helped us recognize that we were Egyptian Jews, and not Jewish Egyptians. This is the essence of Kiddush sanctification - the realization that the Jewish People play a unique role in this world.
The Haggada, the story of our physical exodus from Egypt, is recited over the second cup, symbolizing our physical salvation, which is parallel to "I will save you." A person is a slave to his physical needs. When the people were fed by God in the wilderness, as we are today in a less miraculous manner, they were liberated from the shackles of the physical world in order to concentrate on loftier matters.
Birkat HaMazon, the blessings which remind us that God provides for our sustenance, is recited over the third cup, paralleling "I will redeem you" - the goal of the Exodus was the formation of a unique relationship with God.
Hallel is recited over the fourth cup. Hallel is the praise we bestow on God, recognizing that He said "I will take you to be My nation."
