Although v 8 is a new paragraph, it is linked to the preceding one by the term "obligation." In v 7 Paul calls us to "pay back to everyone your obligations [ὀφειλάς] ...." Then in v 8 he picks up on that term using the verbal form of the same word: "To no one in any way be obligated [ὀφείλετε] except the obligation of loving one another" (my translation). The second occurrence of "obligation" in my translation is not literally present in Greek but it is implied by the neuter article τό. The article is used the way we would use quote marks: "… except the 'to love one another' obligation." The NIV's translation is good: "Let no debt remain outstanding except the continuing debt of loving one another." (I prefer to use "obligation" instead of "debt.")
This statement is then followed by an explanatory clause. "For he who is set on loving the other has fulfilled the Law." I translate ὁ ἀγαπῶν as "he who is set on loving" in order to bring out the meaning of the substantival participle in the present tense, which is being used to identify a person's set character and lifestyle.
Now the two clauses of v 8 are connected by "for" (γάρ). "Let no obligation remain outstanding except the continuing obligation of loving one another, for he who is set on loving the other has fulfilled the Law." The second clause is clearly being brought forward in support of the first in some way. But in what way, exactly? I think the unstated assumption is that the Roman Christians, many of whom were synagogue-attending God-fearers before they came to Christ, would have viewed the Law, especially its moral requirements (with a special emphasis on the ten commandments) as the primary source of their ethical obligations and duties. Therefore, when Paul states that they must let no obligation remain outstanding except the continuing obligation of loving one another, he is essentially taking their focus away from the Law as their source of obligation and telling them to focus on one thing, namely, their neighbor as the source of obligation. But to assure them that they are not going to be missing anything by changing their focus, he says, in effect, if you focus on fulfilling your obligation to your neighbor instead of your self-imposed obligation to the Law, you will have fulfilled the Law as well, even though that was not what you had set out to do.
Thus I would paraphrase v 8 as follows: "Let no obligation remain outstanding except the continuing obligation of loving one another. True, this narrowing of one's duty to love feels dangerously antinomian. By focusing on love instead of the Law, you fear that you will be missing some important obligation or duty required by God in the Law. But there is no need to be afraid, for he who is set on loving the other has fulfilled the Law."
In vv 9-10, Paul provides an argument to prove that this is true. It is a two-step argument. In step one, he singles out four commandments: "You shall not commit adultery," "You shall not murder," "You shall not steal," "You shall not covet." These four commandments all have one thing in common: they are prohibitions against injuring one's neighbor. To commit adultery, to murder, to steal, or to covet, all involve a wrong done to another.
Then he adds, "and if there is any other commandment, it is summed up in the well-known command of Lev 19:18, 'Love your neighbor as yourself.'" The phrase, "if there is any other command," is important, because it covers the concern that the Roman Christians have that they might be missing out on some important duty if they focus exclusively on the continuing obligation of neighbor-love. By implication, yes - let's bite the bullet - there are some other commands, such as the fourth commandment, that are relativized by this. But that's okay with Paul. He doesn't think the fourth commandment is to be kept by literally observing a holy day anyway (see Rom 14:5-6; Gal 4:10). Paul believed that the Sabbath day was a shadow that is fulfilled in Christ (Col 2:16-17). The underlying moral principle contained in the fourth commandment is fulfilled through love, that is, through gathering together with God's people as the body of Christ to worship the risen Lord gathered around his table.
In step one of the argument, then, Paul has made a good case that the Law, both in particular commands and in its summary commands, is aimed at trying to get us not to wrong other people. Paul then brings in the second step of his argument in v 10a: Now we know that "love works no harm to the neighbor." This does not need to be argued for; it's basically part of the definition of "love." If you love your neighbor as yourself, you won't wrong him by sleeping with his wife or murdering him or stealing his possessions. All of this then leads to the logical conclusion stated in v 10b: "Therefore love is the fulfillment of the Law." With that, we are brought full circle back to where we began in v 8b with the thesis that he had set out to prove.
In sum Paul is saying, Make it your aim to let no obligation remain outstanding except the constant obligation of loving other people, of seeking their best interests. Don't set your aim on trying to keep the Law. If your mind is set on keeping rules and commands you will miss the goal. The Pharisees in the Gospels had this mindset. They were so intent on keeping the Sabbath and following all the rules, and yet they lost sight of the weightier matters of the Law - justice, mercy, and faithfulness. Keeping the rules can turn into a self-centered endeavor where the goal is to be able to look at your moral accomplishments and say, "Look, I kept all the rules!"
So Paul says, change your goal. Rather that putting yourself to work on keeping the rules, set your aim on loving others. This may seem too narrow, but it is not. For even though you are not intentionally trying to keep the Law, if you are intent on loving others, if seeking your neighbor's welfare is your set pattern and character, then the fruit of your life will be the kind of fruit that the Law had been trying to get at all along.
Now all of this is quite worrisome, just as it would have been worrisome to the Roman Christians with their background in the synagogue and its comforting ethical boundaries as provided in the Law. I even wonder if Paul himself sensed the need to safeguard this teaching from going off into license, particularly in the area of the sins of the flesh which can easily be rationalized by sinners under the banner of "love." Perhaps this is why Paul goes on in vv 11-14 to say, "And do this – that is, loving your neighbor as yourself – knowing the time," etc. This exhortation places the neighbor-love ethic within an eschatological already/not-yet framework. This is important, because then Paul can exhort us to love our neighbor in a way that involves putting off the shameful works of the flesh which stand judged as part of this present age which is passing away, and putting on Christ and making no provision for the flesh. Of course, the two (love and mortification of selfish sins and desires) are related, because you cannot love others without dying to self.
In other words, with regard to others (vv 8-10), your prime concern and obligation is to love them. With regard to yourself (vv 11-14), your prime obligation is to put on Christ and make no provision for the flesh. If you do this, you're actually going beyond the Law. The Law has no power to curb the flesh and promote walking in the Spirit. By making it your focus to truly love your neighbor, your life will meet the standard required in the Law. If you walk in the Spirit of the eschatological age to come, loving your neighbor, putting to death your flesh in union with Christ in his death, and putting on the risen Christ, you will not only meet but exceed the Law. The fruits of the Spirit in your life will go beyond anything the Law could have ever dreamed of.