I think it's best with this passage to take a few points from it..
1. Paul reminds the Corinthians that he can be called an Apostle, because he has seen the Lord with his own eyes, not whilst Jesus was walking the earth, but, as he terms it elsewhere, 'like an untimely birth' (1 Cor. 15 v 8, Literal version) Jesus met Paul on the road to Damascus.
2. It can be inferred from v. 6 that most of the apostles and church leaders had wives apart from Barnabas and Paul.
3. Paul had an incredibly detailed Biblical knowledge and a great mind with which to apply it. He quotes in verse 9 a passage from the book of Deuteronomy to support his claim that those who teach should be provided for materially.
4. What is Paul's train of thought here? Those Corinthian believers who reckoned themselves mature Christians felt that their special position gave them the right to eat food freely which had been offered to idols and other things. Paul retorts by stating all the things his position should bring him, but yet he refuses to claim in case they turn out to be stumbling blocks to others and hinder the Gospel's efficacy.
5. Despite his arguments with this church, Paul sees them as the guarantee of his apostleship. They were the seal to his right to call himself an apostle, in effect.
6. Paul refused to claim his rightful privileges. Why? He knew all about the priests who had become a byword for luxurious living, because they ate the meat of sacrifices every day. He was determined not to have that accusation levelled against him. He was also independent and perhaps that went too far, as he seemed to have upset the church by refusing all aid.