There are some striking parallels with water baptism and baptism unto Christ. For instance, part of the baptism into Christ involves baptism into Christ's death. Paul even stated how "our old man was crucified with Him", and part of the purpose is so that "the body of sin might be done away with" (6:6).
The crucifixion of the old body is part of the reason why God's redemption plan involved Jesus dying on the cross. In fact, it's part of the "great exchange" done automatically whenever someone believes in Jesus as Lord and Savior. Paul talked about how those who died "has been freed from sin" (6:7). In a literal sense, dead people don't sin, but this sense is also applied when we are baptized into Jesus's death. The old self "died" and thus does not have to remain dominant in life. We know this because doing away with the old self means that "we should no longer be slaves of sin" (6:6).
With these in mind, we also know that we can "be in the likeness of His resurrection" (6:5). A few verses later, Paul will explain what it means, but basically, it's a life where sin doesn't have to be the domineering force.