Owe no one anything except to love one another, for he who loves another has fulfilled the law. For the commandments, "YOU SHALL NOT COMMIT ADULTERY," "YOU SHALL NOT MURDER," "YOU SHALL NOT STEAL," "YOU SHALL NOT BEAR FALSE WITNESS," "YOU SHALL NOT COVET," and if there is any other commandment, are all summed up in this saying, namely, "YOU SHALL LOVE YOUR NEIGHBOR AS YOURSELF." Love does no harm to a neighbor; therefore love is the fulfillment of the law. (Romans 13:8-10)
After commanding Christians to “render to all their due” (Romans 13:7), he uses the discussion about “paying their debts” as the foundation to resume his explanation about Christian love (Romans 12:9-21).
Love is a Perpetual Debt.
Once you pay your taxes (Romans 13:7), that “debt” has been paid or fulfilled. A paid tax bill is one that you no longer owe.
However, there is a “debt” that we continue to “owe” - and that the debt to “love one another.” The Greek word for love that Paul uses in this section is from the word, “agape.” This word means to “have a high regard” for others and “to take interest in their affairs” (Pollard, Truth for Today Commentary, 475).
The one who continued to love his neighbor, “fulfilled the law.”
The context in Romans 13:8-10 indicates that this reference is to the Old Testament - specifically, the Pentateuch, where the citations in 13:9 are found…That a Christian, whether Lawfree or Judaizing, could be said to have fulfilled the Law by loving might have been intended as a message for the Judaizing Christians in Rome. They emphasized observing the Jewish practices of the Law, such as food regulations and observing special days. Paul’s position was that these peripheral matters were not central to Christian belief, but showing love to the brethren was. That was the essence of the Law. - Pollard, Truth for Today Commentary, 475
While a debt like taxes was no longer owed once it was paid, the “debt” to love one another was a debt that could never truly be paid in full.
Love is the Summary of the Law.
Paul cited several commands from the Ten Commandments (Exodus 20:13-15; Deuteronomy 5:17-21) that refer to interactions between people. While these commands were important, they could be “summed up” by the command to “love your neighbor as yourself” (Leviticus 19:18).
Paul was comfortable with the radical idea, fostered first by Jesus (see Matthew 22:37-40; Mark 12:29-31), that nothing a person could do toward fulfilling the demands of the Law was as important as love. These two points are vital for Paul’s discussion in chapter 14 and 15, where he laid out in specific terms what the Christians in Rome must do in order to heal some of the fractures in the church. - Pollard, Truth for Today Commentary, 477
Moving forward from this point into the rest of the “practical section” of Romans, what Paul will teach is grounded in “love your neighbor as yourself.”
The interactions of fellow Christians, and of Christians with the world, must be guided by loving our neighbor as ourselves.
I recently taught a class on the Book of Revelation where I showed The Bible Project's YouTube video entitled 'The Book of Revelation (part 1 of 2).' In it, they make the point that we're to follow Jesus's example of sacrificially loving our enemies, even unto death, just as Jesus did. And that's how we will overcome those who persecute us, and that's how the kingdom of God will be manifested to all nations.