That by not being ridiculously overtly bigoted, they have actually interrogated and rejected their own bigotry. The former is basic and mostly relies on social conditioning. The latter requires reading history and people who are criticizing things with which you may identify and therefore take very personally. The latter is not taught in school and school does not provide the tools (outside of literacy) to do so, so it's a difficult, painful, abd regrettably rare thing to see, usually requiring sone trauma to change.
Going through the process of discovering I was trans and surrounding myself with trans people really made me re-examine how little work I’d done on issues of race, among other things. So many of the little passive aggressive things I found myself getting annoyed at cis people doing, I also found myself doing to people of color. Nothing particularly awful, but definitely inconsiderate.
In this regard I'm probably an ignorant simpleton, so what would be an example of common behavior that people think is fine but is in fact inconsiderate or offensive to others?
100%! And it's structurally ingrained, so it involves a very un-fun process of relearning certain habits that don't feel that bad until you force doses of empathy on yourself, the latter of which I think is in the neighborhood of your experience. It means you have to criticize, forgive, and change yourself, which I personally don't enjoy even though it's so important.