The law has impact on those who are alive. But recall that we were baptized in Christ's death as written in the beginning of Romans 6. So from these two points, Paul talks about how we became "dead to the law through the body of Christ" (7:4).
Paul also describes how "the sinful passions which were aroused by the law were at work [...] to bear fruit to death" (7:5). Some people see the law with intent to break it, but additionally, the law indicates that certain passions are sinful. In other words, it's from the law that we know we screwed up and need a Savior.
After the baptism into Christ's death, Paul talks about how we "may be married to another -- to Him who was raised from the dead" (7:4). This is a figurative use of marriage that can be a reference to the impact on death on the law like in re-marriage, as seen a few verses ago. The purpose of this is that we "should bear fruit to God" (7:4) and "serve in the newness of the Spirit and not in the oldness of the letter" (7:6). So God gave us a calling for our new life.