In order to understand why Paul is making the argument that he is in Romans 6, we must first remember the point he made in Romans 5. Paul has just finished talking about the tremendous love of God which was exhibited in the sacrificial obedience of Jesus Christ. Paul then contrasts and compares the results of Christ's faithfulness with the sinfulness of Adam. It is as if Paul envisions two spheres of existence. The old sphere, the way of life represented by Adam includes sin, unrighteousness, death, and the Law. On the other hand, the sphere of Christ includes righteouesness, life, freedom, and grace. Paul says that one can be transferred from one sphere of existence to the other by the free gift of God because of Christ's faithfulness. Paul even says that God's grace is so abundant that even when sin increased, grace abounded all the more.
With such a dichotomy in place, it is easy to see why Paul anticipates the question that he does in Romans 6:1. "What shall we say then? Are we to continue in sin so that grace may increase?" This is the logical conclusion of opposing the Law to grace. If breaking the Law leads to God providing more grace, then shouldn't we keep sinning so that God can be even more gracious? Paul wishes to make very clear that this is not at all what he intends to say and so he responds "May it never be!". Paul says that even though we have been given grace in spite of our sin we should not continue in sin because the very purpose of that grace was to deliver us from the sphere of sin's power. God has delivered us from slavery to sin through Jesus Christ. It would be foolish to willfully enslave ourselves once again.
Paul uses the imagery of baptism to make this point. He says that when we have been baptized into Christ Jesus we have been baptized into his death. As a result, we are dead to sin. It no longer holds us in its grasp. We are actually free to choose to do good. Why, then, would we ever choose to do evil? Or to put it another way, Paul does not expect Gentile Christians to follow the Law (meaning the Jewish Law regarding things like circumcision, food laws, or Sabbath observance) but neither does he expect them to be lawless. There is now another law which these believers must follow; the law of Christ (which Paul will elaborate on in Romans 8). Christians have been set free from slavery to sin but this is not a freedom to do whatever we want. It is a freedom to submit ourselves in service to another master, the one true Lord. It is freedom to follow the law that brings life rather than the Law that leads to death.