After presenting a terrible dilemma in Romans 7, Paul provides the solution to that dilemma in Romans 8. Paul had painted a terrible picture of the power of sin in the previous chapter. Paul portrayed sin as a force so far reaching in its influence that the human will and God's perfect Law were both powerless to defend against it. Sin, albeit unwilled sin, was the only mode of existence for humanity.
In Romans 8, Paul tells us that there is now another possibility for humanity in Jesus Christ. The law of the Spirit of life in Jesus Christ has set us free from the law of sin and death. This is because God sent his Son in human flesh in order to condemn sin in the flesh. Paul does not elaborate extensively on exactly how this works. He does not explain for us exactly how Christ's death condemns sin. However, it is noteworthy that Paul does not say that God condemns Jesus in the flesh. Paul is not here conveying a picture of an angry God who must punish someone and so punishes his own Son instead of the true sinners, namely ourselves. Rather, Paul says it is sin which is condemned in the flesh. Somehow, sin has met its match in the death of Jesus Christ.
More importantly, Paul says that sin was condemned in the flesh so that the requirement of the Law might be fulfilled in us, who do not walk according to the flesh but according to the Spirit. So in the end, God's good Law is finally upheld (although in a somewhat different way than in the typical Jewish sense since Paul will elsewhere speak against Gentiles keeping the parts of the Law like circumcision, food laws, and Sabbath observance) despite sin's power because the power of the Spirit is greater than the power of sin. Furthermore, just as keeping Israel's Law was supposed to do, the law of Christ leads to peace, righteousness, and ultimately life. This life that is granted by Christ is life here and now in the sense of freedom from sin as well as the hope of future life in the final resurrection.