What does 1 Corinthians 9:5 mean?

Paul has established that he is, in fact, an apostle (1 Corinthians 9:1–2). Now he is establishing that apostles, like other ministers of the gospel, are entitled to certain rights. He places this "right" in comparison to how the Corinthians believe their freedom in Christ allows them to eat food offered to idols and attend functions in idol temples (1 Corinthians 8:1–6).

It's unclear exactly who Paul means in his reference to "brothers of the Lord." Perhaps he means Jesus' actual half-brothers, born to Mary. Or this might mean "brothers" in the same sense as general Christian brotherhood. Or, it might be some other group entirely. In any case, Paul's main point is that he is not claiming his "right" to be supported by those he serves.

Paul uses the term "we" here, which presumably refers to himself and his ministry partner, Barnabas. They have the right to bring along a believing wife on their ministry journeys. He points out that other apostles are doing just that, mentioning the "brothers of the Lord" and Cephas, who is Peter. Presumably, these other apostles may have been receiving financial support for themselves and their wives as they travel from place to place.

Of course, Paul does not have a wife that would need to be supported, based on his teaching about marriage and celibacy in a previous chapter (1 Corinthians 7:6–8). In the context of this discussion, Paul is pointing out that it's reasonable for an apostle to expect support for himself and a wife—and yet Paul is asking for nothing from the believers in Corinth. This is meant to exemplify his teaching about sacrificing one's rights for the sake of others (1 Corinthians 8:7–13).