What does 1 Corinthians 9:14 mean?

Earlier verses gave several arguments for why a minister of the gospel is entitled to financial support from the people he serves (1 Corinthians 9:1–13). Here, in his typical style, Paul reserves the strongest argument for last. Jesus Himself commanded that those who proclaim the gospel should get their living from it. Paul is likely referring to Jesus' instructions when sending His followers out to proclaim God's kingdom during His ministry on earth. Jesus clearly said, in Luke 10:7, "And remain in the same house, eating and drinking what they provide, for the laborer deserves his wages."

Paul's purpose here is to defend the idea that those involved in full-time ministry ought to be supported by the church. His greater point, however, is the idea that this compensation is a right: it's something ministers have every reasonable expectation to receive. That, in turn, is being established in order to contrast with Paul's behavior towards the people of Corinth. He will continue to refuse to take advantage of this command of Jesus. He does not want anyone to miss the gospel because of a misunderstanding about Paul's motives for preaching it (1 Corinthians 9:12–13).