Possible meanings are 1) "we persuade people of the truth of the gospel" or 2) "we persuade people that we are legitimate apostles." (See: [[rc://en/ta/man/translate/figs-explicit]])
Here the word "appearances" refers to outward expressions of things like ability and status. The word "heart" refers to the inward character of a person. AT: "those who praise their own actions, but do not care about what they really are in their inner being" (See: [[rc://en/ta/man/translate/figs-metonymy]])