What does 1 Corinthians 9:2 mean?

Paul is clearly defending his authority, but it's unclear against whom he is speaking. Presumably, some "others" have challenged whether he is a legitimate apostle. Perhaps some groups of Christians disputed whether Paul was in the same category as men like Peter or John. Perhaps Paul is emphasizing that he has not been personally involved with all other churches. Nevertheless, he has been an apostle to the church in Corinth. The church exists because of his ministry there.

Paul describes the Christians in Corinth as the seal of his apostleship in the Lord. Their conversion is the most compelling evidence of the validity of his role as Christ's apostle. Their place in God's family, their growth in the Holy Spirit, prove that Paul is a genuine apostle.

Paul emphasizes this point for a reason: he is going to show he has not been claiming his rights as an apostle, for the good of others. In the same way, he is asking the Corinthians to show love for brothers and sisters in Christ by choosing to not claim their right to eat idol food.