Paul has spent nearly all of the 7 chapters building a case for Christ. Starting with the bad news, that Jew and Gentile are sinners - that we are all sinners before God. That salvation comes through faith in God and not by works. He has used God's promises to Abraham as evidence and shown us that through one man (Adam) sin entered the world and that likewise, through Jesus righteousness comes. Chapter 7 has been a set of questions and answers that Paul has preempted based on his case for Christ.
As we finish chapter 7, from verse 13 we continue looking at the relationship between sin and the Law and righteousness. We have seen that the Law is not bad, in fact, it is holy, righteous and good, but what it does do is show us our sin and that we are dead in our status (our position) with God and our need for salvation.
Verse 13 begins with another question, "did that which is good (the Law) then become death to me?" That is, if I'm dead because of the Law (because it's shown me my status before God), does that mean it's killed me, caused me to die? Again we see the "no, but" response. "Certainly not! Nevertheless".
Remembering that if you are Jewish or a God-fearer (Gentile that knew Jewish law) you had been told that God's Law was our side of the covenant God had with His holy people. That obedience to that Law is everything. Now Paul is showing us the Gospel that Jesus taught and was! That is, we are sinners, unable to adhere to the Law perfectly and therefore await judgement and the wrath of God. To answer the question posed here, Paul says it is not the Law that makes us dead, rather it is our nature.
We miss the mark (God's perfect standard) because we are slaves to sin. We have been sold into slavery. As a result, even though we want to obey God's Law in our mind, we can't fully obey it! Verse 15-20 describes in detail Paul's thought process here. He says, "I do not understand what I do:
What I want to do - I do not do
What I hate to do - I do
Therefore (he continues);
If I do what I do not want to do (which is transgress), I agree that the Law is good.
If I'm not the one in control (slave to sin) it is therefore sin living in me that does it.
This means that because I know what is good, but can't do good, therefore I keep on going evil things. As a result, we can deduce that the Law is good, and that the Law does not put sin in me! This is concluded here:
So I find this law at work: Although I want to do good, evil is right there with me. For in my inner being I delight in God’s law; (Romans 7:21-22)
Paul now introduces another law. We have the Law, which refers to the Mosaic Law (given by God to Moses and the Israelites). In verse 23 we see another law that is working in us (pre-salvation). It is this law that wages war. It is the law of sin at work within me.
This leads us to one conclusion. What a wretched person I am! Stuck between the Law of God and the law of sin. Once we're aware of this situation we are prompted with a follow on question that has eternal consideration:
Who will rescue me from this body that is subject to death?
Who will rescue me? Who will save me? Who can and will pull me from this position, this status before God?
The answer is there in verse 25:
Thanks be to God, who delivers me through Jesus Christ our Lord! (emphasis added)
And through our faith in Christ, His work on the cross, we are presented with a new life. A new life where there is a third law (introduced in Romans 8:1);
...the law of the Spirit who gives life has set you free from the law of sin and death
The Law, holy, righteous and good as it is, can't set us free, only Christ and His spirit can!
Our flesh is weak, and we are unable to create salvation, but the Good News is here. The Good News is Christ.
Romans 8:3, ..."For what the law was powerless to do because it was weakened by the flesh, God did by sending his own Son in the likeness of sinful flesh to be a sin offering." While Jesus took on flesh, he was still perfect and could (and did) act as a sin offering (without blemish, see Numbers 6:14, Leviticus 4:2-4), carrying our sin onto the cross, while covering us in His righteousness, so that verse 4 "in order that the righteous requirement of the law might be fully met in us, who do not live according to the flesh but according to the Spirit, (that means those that are believers). This is the purpose of Christ's work! We are therefore able to meet God's standard, through Christ, not through anything we contribute. We must walk with Jesus every day, walk in the spirit and not by the flesh.