The illustration is of a city state with its own local rulers, or city fathers. In the time of Solomon, cities had their own elders.
So it is in the community of God’s people. Integrity makes a man fit to have responsibility over the church and to govern justly, with gentleness and fairness. This makes the people of God rejoice; it also means that the influence of such leaders will be great, both among God’s people and in witness to a world lost in sin. Our lives will earn us respect even among unbelievers because they know us to be people of integrity and therefore the gospel message will be promoted.