This means that ever admitting you were offensive is a huge status hit implying you are some combination of callous, ignorant, and racist. Sometimes people may be willing to take this status hit, especially if upon reflection they believe they really were in the wrong, but since most people's actions seem reasonable to themselves they will not be willing to accept a narrative where they're the villain.

More likely, they will try to advance an alternative interpretation, in which their actions were not legitimately offensive or in which they have the "right" to take such actions. Such an interpretation may cast the offended party as a villain, trying to gain power and control by pretending to be offended, or unduly restricting the free speech of others.

Sociopathy 101: the best response in this situation is usually to admit wrongdoing. If you try and defend yourself, you'll just dig yourself into a bigger hole. ("I'm not a racist, I just think ... " - we all know how that sounds). You don't need to actually believe you've done wrong, but make it at least sound like you've realised the error of your ways.

You still need to avoid a big status hit, so don't grovel. You should stay "on-message", and your message should be on the lines of "I'm sorry, I didn't realise I was being offensive, but I accept that I was. Thanks for helping me to be less prejudiced. I'm going to try and change in future". How you deliver this message depends on context - if you're not a public figure it's not like you can just hold a press conference, so you'll probably have to deliver this message to individuals, in which case you'll have to make it sound more personalised and natural.

Accepting a small status hit in this way can actually be high status. This strategy also works in the more general situation whenever someone accuses you of being X, where X is some negative trait. Ignore the overwhelming desire to explain why you are not X, with reasons. It will just make it sound like you don't "get it". Even if the criticism is totally invalid, the correct response is to accept it and promise to change.

Exceptions: if you think people will agree that your infraction was minor and the other party is overreacting (especially if they keep throwing new accusations at you after you accept the first), you can (and should) stand up for yourself.

If your infraction was very serious, or you've overused this tactic to the point people realise your tricks, it can backfire badly. I don't really know what to advise you in this situation, but you might need to accept some more-than-token punishment.