In these verses, Paul injects love into the context of law, showing that it is the sum of all duties. He does not say love ends the need for law but that it fulfills—performs or accomplishes—the law.
Notice love's relationship to law in context with what immediately precedes it. The context is a Christian's response to government. He should submit to and honor human government as God's agents in managing human affairs. A Christian is indebted to the government to pay tribute and taxes. When we pay them, a Christian is no longer financially indebted to the state until it imposes taxes the following year.
Regarding men, we are not to be in debt. He is not saying a Christian should never owe anybody money, but that there is a debt we owe to every person that we should strive to pay every day. This debt is one of love, paid by keeping God's law, and this Paul illustrates by quoting several of the Ten Commandments! Inherent in this debt is that no matter how much we pay on it each day, when we wake up the next day, the debt is restored, and we owe just as much as we did the day before!
This sets up an interesting paradox because we owe everyone more than we can ever hope to pay. The paradox, however, is more apparent than real because this is not what Paul is teaching. He is teaching that love must be the driving force, the motivation, of everything we do. This points out a weakness of law regarding righteousness. Law, of and by itself, provides neither enough nor the right motivation for one to keep it.