Romans means that the righteous live by faith because life with God is received by trusting God's saving work, not by establishing one's own worth. Paul uses the line to frame the whole letter: God's righteousness is revealed in the gospel and received by faith.
The phrase closes Romans 1:16-17, where Paul explains why he is not ashamed of the gospel. It prepares for the argument that follows: all are under sin, and justification comes through faith apart from boasting.
This gives Romans its posture of dependence. Faith is not a vague optimism; it is reliance on God's promise and saving action.
Read the phrase as part of Paul's gospel thesis, not as a detached slogan. Romans will spend several chapters showing why faith is necessary and what faith receives.
Original wording grounded in Romans 1, Romans 3, and Romans 4.
17For therein is revealed a righteousness of God from faith to faith. As it is written, "But the righteous shall live by faith."