According to the stereotypes I grew up with in the 50s and 60s, that would refer to good, respectable church-going people who we less virtuous beneath the surface. They might be business cheaters, adulterers or molesters, abusive, or secret drunkards, but they pretended well and kept up appearances. Self-righteous instead of righteous. There was some truth to it, but it was sometimes overblown by people who did not want to be respectable, looking for excuses and people to accuse. That there was social pressure to act this way was also true, though again, I think overblown.
Public righteousness these days is more likely to be enforced from an activist left, and different items are on the righteousness list. In some corners there is a great deal of social pressure, including threat of violence. Fortunately, that isn't true in most places. I don't think those activists picture themselves as the cultural descendants of small-town Baptists of the 1950s, but it's pretty similar, just with more threatening these days.