Why could Paul say the Corinthians were the proof that he was a real apostle?
Paul could point to the Corinthians because they were his work in the Lord: through his ministry they had been saved out of abject idolatry and paganism. Had he not been a true apostle, they would have remained in darkness, so their conversion itself proved his apostleship.
Les explains that Paul asks, “Are not ye my work in the Lord?” and says that if he is not an apostle to others, yet he certainly is to the Corinthians. Paul could say this because he had seen them saved out of abject idolatry only because of his ministry. They were the proof of the pudding: if Paul were not an apostle, they would still be in their paganism. Les applies the same idea, saying that when people question who he is or what business he has teaching the Word, his proof is the many people who have been saved out of especially abject sinful backgrounds and brought out of darkness into the light of the Gospel. That is what Paul is saying: his apostleship is proven by the Corinthians themselves, because he led them out of dark idolatry. The context also notes the complaint from the Jewish community that Paul was not one of the Twelve and had not walked with Jesus three years like Peter.
Am I not an apostle? am I not free? have I not seen Jesus Christ our Lord? are not ye my work in the Lord?
If I be not an apostle unto others, yet doubtless I am to you: for the seal of mine apostleship are ye in the Lord.