"Who serves as a soldier at his own expense? Who plants a vineyard and does not eat of the grapes? Who tends a flock and does not drink of the milk? Do I say this merely from a human point of view?" NIV translation
Paul uses basic human experiences to show how abnormal it would be to work without being paid. We must admit that most of us would not work full time without receiving a paycheck.
"Doesn't the Law say the same thing? For it is written in the Law of Moses: 'Do not muzzle an ox while it is treading out the grain.' Is it about oxen that God is concerned? Surely he says this for us doesn't he? Yes, this was written for us, because when the plowman plows and the thresher threshes, they ought to do so in the hope of sharing in the harvest." NIV translation
Paul backs up the human experiences with the Law as he quotes from Deuteronomy 25. It may seem like Paul is justifying getting paid to do his ministry work but he is not. He is, in fact, saying that those who serve the Lord and receive a paycheck do what is right morally and legally.
"If we have sown spiritual seed among you, is it too much if we reap a material harvest from you? If others have this right of support from you, shouldn't we have it all the more? But we did not use this right. On the contrary, we put up with anything rather than hinder the gospel of Christ." NIV translation
Paul uses logic to show the Corinthian believers that he and his team deserved financial support from the church but goes on to say they did not want that support. We see that the reason that Paul and his team did not take money from the churches was that they wanted the money to be used for other ministry. When he says "hinder the gospel", it is talking about the fact that the "church" only has so much funding and, if they were to demand a paycheck, that money would have to be taken out of some other ministry area such as feeding the widows, etc. Today, there are many "pastors" who are demanding high wages so that they can live in style and they have a right to receive this compensation but they are already receiving their reward for the work that they are doing. Paul was looking forward to his eternal rewards and not focused on the here and now. Those that are receiving their rewards here are also hindering the gospel because their wages could be used to serve others and bring them to faith in Jesus Christ.