This passage is connected to Chapter 8, - can we eat food offered to idols? The answer is yes - the idols are not God. They are made of wood, clay and meal. The food offered remains as food, which can still nourish us. However, not all has this understanding, and if eating food offered to the idols can cause the younger Christians to stumble, then we can give up that right to eat the food, for the sake of our fellow Christians.
We have rights but we should use it for the benefit of others and for the glory of Christ. Apostle Paul was explaining his rights to question the church of the Corinthians for he founded the church.
"Mine answer to them that do examine me is this, Have we not power to eat and to drink? Have we not power to lead about a sister, a wife, as well as other apostles, and as the brethren of the Lord, and Cephas? Or I only and Barnabas, have not we power to forbear working?" 1 Cor 9:3-6. Apostle Paul stated his rights like any other - to eat, drink; to marry; to work. However he chose not to, because he wanted to dedicate his life for Christ and the gospel. This is an example that we need not always exert our rights all the time.
"If we have sown unto you spiritual things, is it a great thing if we shall reap your carnal things? If others be partakers of this power over you, are not we rather? Nevertheless we have not used this power; but suffer all things, lest we should hinder the gospel of Christ." 1 Cor 9:11:12. Apostle Paul stated that he have the right to ask for payment for his ministry. Deuteronomy 18:1-5 also stated how the Levits should be paid by the people. However Apostle Paul was saying that he did not want any payment because he wanted the Gospel to be made freely to all, without obligations for payment.
"yea, woe is unto me, if I preach not the gospel!" 1 Cor 9:18. Apostle Paul will give up all rights to preach the gospel. That's his focus and purpose. We will do well to learn from him.