We can infer from the letters of Paul that some people questioned his apostleship, which might be part of the reasons why Paul introduced himself this way in his letters. He had to prove it over and over again that God actually made him an apostle (2 Corinthians 12:12).
Paul didn’t say it because he wanted his readers to see how special he was or to show that he was above them. He wasn’t trying to glorify himself, but he said it out of necessity. So that the people would believe that the message he preached is not of human origin but was committed to him by our lord Jesus.
Why people questioned his apostleship might be because he never saw Jesus during Jesus’ earthly ministry and that alone disqualifies him from being an apostle based on the criterion Peter gave while they were planning to replace Judas Iscariot (Acts 1:21-22). So some of the Christians especially the Jews probably felt Paul wasn’t qualified to be called an apostle. What they didn’t know is the fact that no one can be qualified for it and for as many that God has called to be an apostle of Christ, none has ever deserved it; they were all picked by grace.
Really, what they were doing was questioning if God actually called Paul into the ministry. Therefore, Paul declared that his works are his proofs. I mean we couldn’t have demanded for more, concerning the works expected from an apostle!
I have seen people that really have issues with people calling themselves some religious titles like pastor, evangelist, prophet or apostle. I really think that should not be a problem because bearing it doesn’t show pride and not bearing it doesn’t depict humility. The fact that Paul called himself an apostle of Jesus means it might not be wrong for someone who is called into that office to address himself as such. But as far as I am concerned, I think religious offices are job descriptions for individuals partaking in the grace of Christ and might not be used like secular or political titles but if anyone thinks he needs to use it as a title, I don’t think such has committed a sin. It shouldn’t be a source of division or disagreement among believers.
I really don’t regard spiritual leaders by the tittle they bear, but by the works they have done; how they have contributed to the growth of the ministry of Christ on earth. And really we could see this in Paul, not only was he just saying it around that he was an apostle, he did great works that cannot be over-emphasized. He really was an apostle of Jesus Christ.