What does 1 Corinthians 9:6 mean?

Paul has been asking a series of rhetorical questions. These are not actual attempts to learn, but statements that expect a certain answer. His point is that he has not been demanding his "rights" as an apostle (1 Corinthians 9:1–2). Other apostles, he has suggested, receive financial support from those they serve, even bringing their wives along on ministry travels. Now Paul implies that he and Barnabas also have the right to ask for this support, rather than working an outside job to support their own ministry efforts.

Paul's reason for using this style of writing is to highlight the obvious: of course they have the same right. Paul and Barnabas are willingly choosing not to take money from the people they minister to.

Paul's mention of Cephas/Peter in the previous verse, and Barnabas here, shows the Corinthians at least know who these men are. It's not clear if either ever came to Corinth. Paul and Barnabas traveled together earlier in their career, perhaps agreeing during that time not to take funds from the people they evangelized.