What does it mean that love is the fulfillment of the law in Romans 13:8?

TL;DR:

The love that fulfills the law is love that reflects God's character and His attributes. When we love like Jesus, we reflect and reveal Him to the world.

Romans 13:8 says, "Owe no one anything, except to love each other, for the one who loves another has fulfilled the law." In what way is love the fulfillment of the law? In short, love is the fulfillment of the Law in that all of the Old Testament law can be expressed by the commands to love God and love others (Matthew 22:36–40). Love fulfills and expresses the intent of the Law. Jesus also commanded us to love (John 13:34–35; 1 John 3:11). When we love others, we are doing the will of God. As a point of clarification, it must be stated that this love is the love of God: a self-sacrificial love aimed at the good of others. It is not love the way the world thinks of love. When we love as God has called us to, we reflect Him to the world.

from the old testament

The Ten Commandments and the rest of the Mosaic covenant was given to Moses on Mount Sinai after the exodus from Egypt (Exodus 19—24). The Old Testament law reflects God's character, and it is summed up in love.

from the new testament

Romans 13:8 says, "Owe no one anything, except to love each other, for the one who loves another has fulfilled the law."
Jesus affirmed love as the essence of the Law when He replied to a Pharisee's question about the greatest commandment. Jesus said, "You shall love the Lord your God with all your heart and with all your soul and with all your mind. This is the great and first commandment. And a second is like it: You shall love your neighbor as yourself. On these two commandments depend all the Law and the Prophets" (Matthew 22:37–40).
Jesus explained to His disciples, "This is my commandment, that you love one another as I have loved you. Greater love has no one than this, that someone lay down his life for his friends. You are my friends if you do what I command you" (John 15:12–14).
Jesus told His disciples, "A new commandment I give to you, that you love one another: just as I have loved you, you also are to love one another. By this all people will know that you are my disciples, if you have love for one another" (John 13:34–35).
Jesus is the perfect example of love. When we love like He did, we demonstrate Him to the world. We who have been made children of God through the new covenant (John 1:12; Galatians 4:4–7) should reflect our Father, and that is done through love (1 John 4:7–12).
Romans goes on to explain: "For the commandments, 'You shall not commit adultery, You shall not murder, You shall not steal, You shall not covet,' and any other commandment, are summed up in this word: 'You shall love your neighbor as yourself.' Love does no wrong to a neighbor; therefore love is the fulfilling of the law" (Romans 13:9–10).
Paul says something similar in Galatians 5:14: "For the whole law is fulfilled in one word: 'You shall love your neighbor as yourself.'" In short, the Old Testament law was aimed at love. To love others, then, would be to fulfill the Law. Many of the specific laws detailed what loving others looks like in practice.
First John 4:7–12 explains, "Beloved, let us love one another, for love is from God, and whoever loves has been born of God and knows God. Anyone who does not love does not know God, because God is love. In this the love of God was made manifest among us, that God sent his only Son into the world, so that we might live through him. In this is love, not that we have loved God but that he loved us and sent his Son to be the propitiation for our sins. Beloved, if God so loved us, we also ought to love one another. No one has ever seen God; if we love one another, God abides in us and his love is perfected in us."

implications for today

We are saved by God's grace through faith (Ephesians 2:1–10), not through any law nor through our ability to love others. When we are saved, we are made a new creation (2 Corinthians 5:17). We receive the indwelling Holy Spirit who works to transform us to be more like God (Ephesians 1:3–14; Philippians 1:6; 2:12–13; Romans 8:28–30). We have freedom in Christ and are not bound to the Old Testament law, but God still has expectations for His followers. We obey Him out of our love for Him and because of the changes He has made in our hearts (John 15:1–17). We are freed from sin and to righteousness (Romans 6:15–23). The more we know God, the more we reflect Him, and the more we truly love and live.

understand

The Law can be summed up by loving God and loving others.
Love fulfills the Law and sums up obedience to God
When we love, we reveal the heart and character of God.

reflect

How does understanding that love is the fulfillment of the law change your perspective on obedience and living according to God’s will?
How does loving others as God loves us impact your relationships and interactions with people in your life?
How does the self-sacrificial love of Jesus serve as an example for us in fulfilling the law through love?

engage

Loving God and loving others essentially sums up the Mosaic law. Thus, love is the fulfillment of the Law. Jesus fulfilled the Old Testament law in its entirety (Matthew 5:17–18). He lived a perfect life, died on the cross as a payment for our sins, and rose to life victorious over sin and death. He made the old covenant obsolete and instituted a new and better covenant (Hebrews 8:1—10:25). But love is still God's command for His followers. How does Jesus’ command to love others as He loved us reflect the depth and sacrificial nature of God's love?
In Greek, love in Romans 13:8 is from the word agape. Many cultures today use the word "love" to mean a variety of different things, including affirming others in their rejection of God. But that is not loving in any way. Genuine love is rooted in the truth of who God is (Ephesians 4:15–16). God’s character and attributes are reflected through the Law, including His mercy, grace, holiness, and justice. If "love" is not rooted in the truth of God, it is not the love of God nor the love to which God calls His followers. How does the concept of love as the fulfillment of the Law differ from the common cultural understanding of love? What implications does this have for how we define and express love today?
What does it mean for us practically to live out love as the fulfillment of the law in today’s world? How does this look in different areas of life such as family, community, work, and church?