In these verses we find Paul preempting a question that might be asked following the statement made in the chapter before:
"But where sin increased, grace increased all the more" (Romans 5:20)
What's the problem here, and what is the idea he attempts to address in this next chapter?
The problem is cheap grace. The idea, that if I am saved through faith, and my sins (past, present and future) have been forgiven through the free gift that is Christ on the cross, that I, therefore, do not need to be obedient to God. As my sin increases, so does God's grace.
What we are provided with is an illustration of slavery. Something very familiar in 1st Century Rome, and still, unfortunately, something we have in the world. The description that we are no longer slaves to sin (v6), for sin shall no longer be your master (v14). In the chapters before we have learned about justification and reconciliation, now we explore the idea of sanctification.
The idea that we are caught "in limbo". Our eternal life is secure, but we await the day of our final salvation. We are 'died to sin', linked to the death of Jesus and the shift in our state, our new position before God. With this changed position before God, comes the realisation that we can no longer live as we did. That we can no longer live in sin as we use to. We are told that all of us (believers) have been baptised into Christ. We have joined Christ in His death and His resurrection. Baptism is used because it shows and reminds us of the major events of his redemptive work. Died, buried and risen - with Him! Through this, we now have the ability to live a new life, in Christ and for Christ.
For if we have been united with him in a death like his, we will certainly also be united with him in a resurrection like his. (Roamsn 6:5)
We are citizens of heaven, but we are not there yet! We eagerly await our Saviour's return. However, during this period of 'limbo', we are no longer slaves to sin (v6), rather we have been set free from sin (v7). Verse 8 shows us that 'living with Christ' automatically follows dying with Him. This means that even though our bodily resurrection is in the future, we enjoy the benefit of Christ's resurrection. Verse 9 shows us that we also have victory over death. Tells us that even Jesus was subject to the power of sin, even though He never sinned.
Since the children have flesh and blood, he too shared in their humanity so that by his death he might break the power of him who holds the power of death—that is, the devil— and free those who all their lives were held in slavery by their fear of death. For surely it is not angels he helps, but Abraham’s descendants. For this reason he had to be made like them,[a] fully human in every way, in order that he might become a merciful and faithful high priest in service to God, and that he might make atonement for the sins of the people. (Hebrews 2:14-17)
Reading Romans 6:11-12, we see an imperative after the indicative. That is, we are dead to sin but alive to God in Christ Jesus, therefore do not let sin reign. Sin is no longer our master, yet it does not give up trying. This is not an automatic process, or valve you can simply stop, or close off. Rather we have been "called out" of our old life. Verse 13, rather we are to offer ourselves to God.
Taking an illustration from Martin Lloyd-Jones, we can think of two fields. In the first field, there is sin. This first field is governed by Satan. This is where we were before we believed, in the darkness, trapped and bound by sin. However, when we believe in Jesus, we are then taken from that field by God and placed in the second field. One of righteousness, one where Jesus rules! In between these fields is a large fence we can not climb ourselves, we must rely on God to lift us out. But once we are in that second field, by God's grace through faith, we can still hear and see the first field. Our task (with the power of the Spirit) is to move away from the fence - further and further away from the first field. We're still in the second field, but as we move (mature) away from the fence, the call of sin will be quieter and quieter. That is sanctification!