Jesus in the gospel of Matthew:

 “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 22:36-40)

Paul in his letter to the Romans:

Owe no one anything, except to love each other, for the one 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 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 13:8-10)

I have occasionally heard Christians say something like “Keeping the Law doesn’t matter; what matters is loving people.” I don’t think that’s what Jesus and Paul mean here. If they did, it’s unlikely Jesus would say this earlier in Matthew’s gospel:

“Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them. For truly, I say to you, until heaven and earth pass away, not an iota, not a dot, will pass from the Law until all is accomplished. Therefore whoever relaxes one of the least of these commandments and teaches others to do the same will be called least in the kingdom of heaven, but whoever does them and teaches them will be called great in the kingdom of heaven. For I tell you, unless your righteousness exceeds that of the scribes and Pharisees, you will never enter the kingdom of heaven.”
(Matthew 5:17-20)

Neither Jesus nor Paul thought the Law was overbearing or unnecessary. So when they say that loving God and neighbor is the fulfillment of the Law, I think they mean that “Love God and love your neighbor” is an accurate summary of the Law; it expresses the heart behind the Law.

Does this render the Old Testament law unnecessary? Not at all.

In a perfect world filled with perfect people, God could say “love me, and love one another,” and that would work. But we’re sinful people living in a broken world, so that isn’t enough. We need to know how to love God; we need to be told how to love one another. That’s what the Law does. It describes exactly what God expects from his people. And since his people are sinful, it graciously gives provisions for how we repent when we fail to keep his law.

Another dimension to the Law: One feature of our sinfulness is that we tend to think of ourselves more highly than we ought. So rather than leave it at “love God and love one another,” knowing that we might well respond like the rich young ruler and say “All these I have kept from my youth,” God shows us how high the bar is. You say you love me? Have you ever taken my name in vain? Have you kept a weekly Sabbath of rest and worship for your entire life? You say you love others? Have you ever told a lie about someone else? Have you ever been jealous of someone else’s property?  The Law shows that we can’t keep the Law. It makes us want to look for a Savior. That’s by design.

Far from relaxing the Law to a broad we could claim to have kept, Jesus and Paul are showing us what the Law is about. It’s about loving God and loving our neighbor. We can’t do either of those, which is why Jesus came. His life and death covers over our failures, and frees us to pursue obedience with a renewed mind and heart.