The Purpose of an Annulment is to Receive a Cleansing
Many people believe that a divorce is the by-all-end-all to a marriage. A civil divorce is only legal in the eyes of man's law and government. In the eyes of God, a man and woman are married for life when they stand before Him and say "I do." Thus, grounds such as "irreconcilable differences" don't exist for God.
Thus, if a Catholic were to obtain a civil divorce and not a Catholic annulment, they would be committing adultery in the eyes of the Lord. By obtaining a Catholic annulment, you are, in essence, saving yourself from sin and receiving a cleansing in God's eyes. This is also why you are allowed to remarry because you are considered pure once again as you were before your first marriage, i.e. a virgin.