Romans 13:6-7 meaning

Paul gives an example of a law to obey: For because of this you also pay taxes (v 6). We should pay taxes. Remember, rulers are servants of God (v 6), they are acting on God's behalf devoting themselves to this very thing (v 6), as justices and lawmakers and peacekeepers. With the perspective that God is above our government, we should obey the law. We should pay our taxes and show respect to officers of the law, judges, and elected officials.

Romans is about living righteously by faith. If we are law-abiding, tax-paying, well-behaved citizens, we are living as God designed for us to live, obeying Him by obeying our governing authorities, and living in harmony with others. We are to render to all what is due them: tax to whom tax is due; custom to whom custom; fear to whom fear; honor to whom honor (v 7). 

A custom here likely refers to a toll or tariff. The term fear is in context of an authority—caring what they think and giving heed to the consequences they can bring to bear upon you. We are supposed to recognize authorities that can create consequences for us and pay them appropriate heed. The Greek word translated honor can also be rendered "price." The idea seems to be to value authorities, to give them appropriate attention. 

Respecting the law and legal authority is part of bringing righteousness to a community, state or nation. Righteousness means harmony with others in our community. The Greek word translated "righteousness" in Romans, "dikaiosune," is also properly translated "justice." Justice occurs when everything in a community is lined up with laws that support freedom and harmony. The government's job is to address those who are creating disharmony, such as robbers and murderers, and prevent their actions from disrupting citizens' ability to live together constructively.

When God set up the nation of Israel, He established it as a self-governing nation under the rule of law. He entered a covenant (or treaty) with Israel, which the people of Israel voluntarily agreed to (Exodus 19:8). The covenant contained His laws, which Jesus summed up as loving and giving respect to God as the ultimate authority and loving our neighbor as ourselves (Matthew 22:37-39). When we walk in the Spirit we fulfill the Law (Romans 8:4).