To redeem them that were under the law, that we might receive the adoption of sons.
Note 8 at Ga 4:5: When the right time came (see note 4 at Ga 4:4), God sent forth His Son in His preexistent state from heaven into the world. He was born of a woman (virgin birth) and was made subject to the requirements of the Law (see note 7 at Ga 4:4).
His purpose in coming into the world was twofold. First, He came to "redeem them that were under the law." He did this by perfectly keeping the Law, fulfilling it, and paying its curse (Mt 5:17 and Ga 3:13). Thus Christ delivered us from the entire system of the Law. The Greek word for "redeem" used here is "EXAGORAZO," and according to Wuest's Word Studies from the Greek New Testament, it means "to buy out of the slave market" (emphasis mine). Wuest goes on to say that "the word law is not preceded by the definite article, hence law in general is referred to here" (Volume 1, p. 115). This system of Law was then superseded by grace, with an emphasis on living by faith.
Second, Christ gave us the status of son-ship with all its privileges. In the Greek, the expression means adult-sons. Under grace, we are treated as adults, not babies.
Therefore, we were redeemed not only from the bondage of the Law but also unto son-ship. Many people stop short, only realizing what we were delivered from. Sure, we need to rejoice in that, but we also need to move on into realizing the full benefits of our inheritance (Ro 8:19, see note 10 at Ga 4:7).