The Gospel is offensive because it is free, not transactional.
The Corinthian Christians looked upon Paul with askance, harboring doubts regarding his status as an apostle, because he refused to accept their support (patronage) and instead labored to support himself. In their cultural beliefs and traditions, teachers needed to free themselves from labor so they could have time to cultivate their minds. Teachers were expected to be clients of wealthy patrons. “You give us teaching; we support you.” That was the expected transactional relationship.
… There was no doubt friction between Paul and the Corinthians because he worked with his hands. For most Jewish believers this would not have been a problem… The rabbis supported themselves financially, often through some trade or skill. Indeed they were required to do so… On the other hand, for Christians with a Greek background things were different. Intellectuals were expected to be financially independent. Only with the leisure that comes from such independence was it possible to cultivate the mind. How could Greeks accept the intellectual and spiritual leadership of a tentmaker? 
But Paul refused patronage. Why?
First Corinthians 9:1-18 can be seen in two parts. In the first part Paul vigorously defends his status as an apostle and defends rights to which he is entitles as an apostle. Paul must first establish that he knows full well that he is entitled to the support of the Corinthian Christians. In the second part Paul vigorously defends his reasons for refusing patronage.
Paul's response to this is not first of all to defend his renunciation of his rights, but to establish that he has such rights. This must be done because they have questioned his authority altogether. From their point of view his activity would not have been the renunciation of assumed rights; rather, he must have worked with his hands because he lacked such rights. Since, therefore, he did not do as the others—accept patronage—he must not be a genuine apostle. 
One of the foundational problems Paul saw in the Corinthian church was that of freedom and rights. The Corinthians had a distorted view of Christian freedom. Not only did they appear to believe that this freedom meant they could do anything, they were flaunting it and imposing it on those who did not believe this way or were uncomfortable with this particular expression. In other words, some in the Corinthian church were (ironically) becoming enslaved to freedom. This is the point at which 1 Cor. 9:1-18 ties into the issue of food offered to idols that Paul had touched on in chapter 8.
Some of the Corinthian Christians had come to believe that accepting the gospel and its freedoms meant they were obligated to perform and behave in ways that would highlight their freedom and rights. They were still thinking in transactional terms.
By refusing patronage, Paul holds himself up as an example of genuine Christian freedom. He illustrates the transaction-free nature of the gospel through his actions.
Like the Corinthian Christians, we usually don’t have too much problem understanding and accepting that there is nothing we can do to merit God’s grace to us. But we often have a real problem in accepting that there is nothing God demands from us in return. (This is different from God desiring much from us.) There is nothing we can “do for God” for us to keep his grace.
This is why the gospel is so offensive. It is not so much the content of the gospel as it is its very nature. In the honor-shame culture in which this was written, to accept a gift without strings attached, without demand or expectation of reciprocation, was to suffer shame. The gospel is a stumbling block, it is shameful, because its recipients must acknowledge their shamefulness. There is no personal honor in accepting the gospel.
The parable of the tax collector and the Pharisee (Luke 18:9-14) brings home this point. In this parable Jesus says that the tax collector is the one who was justified. The tax collector receives God’s grace. He, unlike the Pharisee, offers no good deeds as an offering. The tax collector offers no promises of sacrifice, offerings, or to even change his ways. He only acknowledges the reality of his condition. That is all grace requires.
The Corinthians wanted to bring transaction back into the gospel so that it could be used as a means of gaining personal honor.
Paul would have none of that. He would not place himself back into the slavery of the honor-shame system. And he warned the Corinthians of the path they were on. Genuine Christian freedom is found in rejecting the relational systems of this world and allowing only Christ to have a claim on one’s life.
In one sense his "pay" is in fact to receive "no pay"! But in the present argument this nonpayment "payment" also gives him his apostolic "freedom" from all, so that he might the more freely make himself a slave to everyone (v. 19). Thus in terms of his own ministry, his "pay" turns out to be his total freedom from all merely human impositions on his ministry. 
Freedom found in giving up one’s personal rights.
 Bailey, location 2874.
 NICNT, introductory text for 1 Cor. 9:3-14.
 NICNT, entry for 1 Cor. 9:18.