Paul begins this chapter by reminding the Corinthians that, as a minister of the gospel and an apostle he is entitled to receive financial/material support from them. Although a bit crude, not muzzling the ox that treads the grain, is a marvelous image for pastoral support. From previous comments it is unlikely that Paul is a married man (in previous chapters he has argued that people should be as he is -- unencumbered and free from marriage responsibilities) however, he does hold on to the "right to travel" with a believing wife. Notice that the other apostles, the Lord's brothers and Peter (Cephas) all are married and all travel with their spouses. This and other passages establishes the Christian tradition of paid clergy. There are exceptions, curiously Paul is one of the exceptions as, at least in Corinth anyway, he chose to work as a maker of tents rather than be a burden on the community.
The principles laid down in 19 and following are vital for understanding Christian ministry and evangelism. Notice that Paul does not expect the rest of the world to conform to his expectations and standards. Rather he is willing to adopt the mannerisms and expectations of various segments of society in order to communicate the gospel to them. When he is among the Jews he plays up his jewishness, when he is among the gentiles he plays up his connections with them. This is why he uses two names -- he is called Saul (his Hebrew name) when he is working in and around Palestine but he calls himself Paulus (Paul -- his Greek name) when he is out among the gentile world.
Apostles were first century missionaries and church planters. The Church planter must understand the culture in which the church is being planted. There are different expressions, different themes, different cultural expectations from community to community, country to country, culture to culture. The Apostle has to be recognized and have understanding before he/she can hope to be heard and to be understood. In a sense the Apostle has to be a cultural anthropologist as well as a preacher of the Gospel. Several places in Paul's ministry we see this playing out. When he is in Athens (Acts 17), although he is grieved by the idolatry of the city he does not begin by renouncing or denouncing the cities idolatry. Rather he uses the multiple altars and the environment as a beginning point for his preaching of the gospel.
How could the 21st century Christian church be better cultural anthropologists so that we may more effectively communicate the gospel?