Paul was continually embroiled in conflict with Judaizing teachers, who he here refers to as “false brethren.” Their view of the Christian faith nullified the grace of God. They viewed Christ as continuing the covenant between Jews and God, and viewed any Gentile Christian as a proselyte.
Circumcision, as a token of the covenant between God and Israel became the point of contention, and Paul was not going to compromise the gospel to appease these evil men. He protected Titus (a Gentile) by refusing to allow him to be circumcised. Paul’s correct understanding of the covenant of grace was that both He (a Jew), and Titus (a Gentile) were spiritual Jews, whose circumcision was not of the flesh, but of the heart! (cf. Romans 2:28-29).
We have liberty in Christ. Our appeal is to God’s grace. An appeal to the Old Law brings men into the bondage of sin.