He tells the Corinthians that he isn't like these peddlers of God's word. He doesn't need to carry around a bunch of testimonial letters that tell everyone how great he is. Oh, so now Paul doesn't like letters?
The proof is in the pudding for Paul. After all, he founded a whole church in Corinth. That's all the letter he needs. Boo-ya.
Paul goes on to say that his letter (the church) isn't written in ink or chiseled on stone tablets. It's written by the Spirit of God on people's hearts. Awww.
He Fought the Law
Don't get Paul wrong. He's not claiming to know any of this himself. God told him.
God revealed a new way of doing things with Jesus.
Now people don't have to worry about obeying every single letter of Jewish law. They just follow God's Spirit in their hearts.
Paul retells the story of the time Moses put a veil over his face (Exodus 34:33-35) to show that those who continue to follow the law have hidden themselves from the truth. They're covered. They're in darkness. Those are some pretty thick veils.
But Christians have lifted the veil and see God for who he really is. Note to Corinthians: throw away all veils.