Adam Clarke on Galatians 4:5

To redeem them - Εξαγορασῃ· To pay down a price for them, and thus buy them off from the necessity of observing circumcision, offering brute sacrifices, performing different ablutions, etc., etc.

That we might receive the adoption of sons - Which adoption we could not obtain by the law; for it is the Gospel only that puts us among the children, and gives us a place in the heavenly family. On the nature of adoption see the notes on Rom 8:15.