Paul was telling the Corinthians to regard himself, Apollos and Peter as servants of Christ and as stewards of the mysteries of God. This tells us a lot about how Paul saw how his, and other people's, relationships to God ought to be. While some might look at being a apostle as a high office, Paul saw it as an opportunity to serve. We need to be looking for opportunities to serve to. (John 13:1-17) Also think of the parable of the talents (Matt 25:14-30)
Paul talks about the mysteries of God here. See what he says in Ephesians 3:1-7. The Gospel is a mystery that needs to be handled with great care by the stewards of God (Col 1:24-27).
Paul lived his life with a good conscience (Acts 23:1). Yes, he was wrong at times, but he was true to what he knew. Paul understood that there would be a day when he would be judged (Rom 2:14-16, Rom 14:10-12). It will be when the Lord comes again (Matt 25:30-33).
Paul gives a warning about not exceeding what is written. The context of this warning seems to do with how they were viewing Paul and Apollos, but of course it has much wider applications.
What we have is not something to boast of. We have nothing that we have not received, and these we have received from God (John 3:27, James 1:17). It is very easy to feel puffed up and proud.
Job 1:21 And he said, “Naked I came from my mother’s womb, And naked I shall return there. The LORD gave and the LORD has taken away. Blessed be the name of the LORD.” (Eccl. 5:15).
When Paul looked at his life as an apostle he could see nothing that those of the world (worldly) would find to boast in. But that is how it ought to be. He was a servant, sent to serve. What he had, he had from the Lord. He gave God the honour and sought none for himself.
Paul claims a special relationship with the Corinthians. They had many teaches – Apollos was one (1 Cor. 3:6), but he had brought Christ to them in the first place and as such he was their “father through the Gospel”.
A note here about Paul’s teaching (Vs. 17). He taught the same thing everywhere, in every church. Many will dismiss what Paul is saying in one letter, saying that there were special circumstances. However, Paul taught the same, everywhere.