Twain doesn’t reveal what, exactly, Reverend Burgess did to lose the town’s respect, but the specifics of this backstory aren’t all that important. What
is important is the way Edward talks about Burgess. Indeed, the fact that he didn’t help clear Burgess’s name even though he (Edward) was “the only man who knew [Burgess] was innocent” suggests that his desire to uphold his reputation eclipses his morality. It also suggests that Burgess may have been falsely accused, and that Edward was the one who actually deserved the town’s scorn. Of course, it’s impossible to determine this for sure, since Twain doesn’t give further details about this incident, but it’s reasonable to wonder how Edward knows Burgess is innocent in the first place.