"The high priest carries the blood of animals into the Most Holy Place as a sin offering, but the bodies are burned outside the camp. And so Jesus also suffered outside the city gate to make the people holy through his own blood. Let us, then, go to him outside the camp, bearing the disgrace he bore." NIV translation
This loyal love is loyalty to Jesus Christ and not the ways of men. Paul urges the Hebrew believers to accept the "shame" of being a Christian. They were trying to be accepted by the Jews at the same time that they were trying to follow Christ. Paul reminds them that Jesus was rejected by the Jewish religious rulers and He said that the rulers would hate them as well. The same thing applies to us who did not come out of a Jewish background. We are called to follow Jesus and not a religious system. If and when we decide to do that, the religious establishment will hate us just like they hated Jesus. Paul reminds the Hebrews that they must go to Christ outside the camp and we must as well.
Today, there are those who want to bring a lot of the Jewish practices into the Christian faith but they do not fit. As a Christian, we trust in the finished work of Jesus Christ to save us but that is incompatible with the idea of following a religious system to be saved.
There are even many today that will tell Jews that reject Jesus as the Messiah that they will get another chance to accept Him after they die. That, my friends, is a lie that is straight out of the depths of hell. Telling people what they want to hear instead of the truth of the gospel is not loyal love (hesed). Real hesed is loving them enough to tell them that they must accept Jesus Christ or they will be condemned. It is not popular but it is the clear truth that is contained in the gospel.
"For here we do not have an enduring city, but we are looking for the city that is to come." NIV translation
The city of Jerusalem is probably the most controversial piece of land in the world. Many forget that there will be a new Jerusalem and that is the one that is most important.