"Owe no man any thing, but to love one another" means that we have to pay our bills, do our job, pay our debts, do our chores at home, do what we said we would do, etc. We have to do what we are supposed to do. And when we owe nothing to anyone, we owe them one thing, which is to love them. Because the Bible says in Romans 10:9: "Thou shalt love thy neighbour as thyself"
Once we fulfill our obligations the only thing that we owe is to love one another.