The Jews considered circumcising on the Sabbath a lawful Sabbath activity. Why? The Bible does not give a direct answer. It is in this point that Jesus nailed them to the wall! The Jews knew why circumcision was lawful on the Sabbath: It was a redemptive act because circumcision was an Israelite lad's introduction to entering the covenant. So circumcision was a redemptive act, even as today we consider baptism a redemptive act. And we rightly, lawfully, will baptize people on the Sabbath.
The Jews' reasoning was that it is lawful and right to cut off a piece of skin from one of the 248 (by their count) parts of the body to save the whole man by making this person a part of the covenant. Christ's reasoning, then, was that works of salvation are accomplished, not only by the Father, but also by His servants, who are His agents. In this case, the priests did the work of circumcision. And the Jews considered it lawful.
Jesus' reasoning is beautiful: "If you can do this act to save a man, why can't I also make a person whole and save his physical life on the Sabbath?" He says, "This is the work of God." It is redeeming somebody, setting them free, giving them liberty.
For Christ, the Sabbath is the day to work for the salvation of the whole person, physically and spiritually. If it is legal to cut off a part of a boy's body on the Sabbath because of the covenant, they have no reason to be angry with Him for mercifully restoring a person to wholeness. His opponents, however, cannot perceive this. It somehow does not enter their minds. We can understand why: God just was not working with them yet - it was almost as if they had blinders on. They could not perceive the saving nature of His work. To them, the pallet (John 5:8) and the clay (John 9:15) were more important than the healed man himself.
John W. Ritenbaugh
The Fourth Commandment (Part 3)