Sometimes, no matter what you do, you're going to offend people. Sometimes you'll never know why.
Things I have done in the past week that someone has found offensive: listened to loud, "weird" music. Had an opinion about whether or not people who aren't me should be allowed to make decisions about my body. Enjoyed bad science fiction. Had my hair highlighted in preparation for the Hugos. Implied that there's a double standard in how women are expected to dress for the Hugos vs. how men are expected to dress for the Hugos. Implied that it's more expensive to be female. Bought children's toys for myself. Bought children's toys for a child. These are just the things I know about mind you, and I only know because in each case, someone told me. I'm not sure why most of these things were offensive. I don't actually want to know. And that, right there, probably offends someone.
I do my best to Marilyn Munster my way through life, leaving fields of happy zombies and sparkly plagues behind me. Sadly, though, nothing is that inoffensive. Not unless it's, say, a rock, and even that will offend, if it gets into somebody's shoe. There is no way to avoid giving offense. Not if you're a thing that actually exists.
And it can be hard, as someone whose audience is largely online, to deal with the thought that I might accidentally offend someone, lose potential readers, and wind up living in a cardboard box next to the creek. My cats aren't supposed to go outside! (This is the "worst case scenario" mindset. It kicks in when I think I've upset someone. My brain is a theme park that hates me.)
This is something I need to remember, myself, particularly when I get hit by one of those "I can tell what the person found offensive, but I can't for the life of me tell what they find offensive about it" moments. Or even better, the "someone is terribly offended at something I didn't say but they think I did," which tends to stick in my craw even worse.
The other thing I've learned the hard way, is there's no point in trying to defend yourself. People dig in and just go after you all the harder. All you can say is "I'm sorry to have offended you" and bow out, regardless of whether you think it's a fair reaction or not.
(Once, when I expressed the opinion that an anthology of "LGBT zombie stories" was a goofy idea, I got all kinds of vitriol from people claiming I hated LGBT folks and wanted to keep them out of literature. Every attempt I made to point out that I hadn't said that, didn't believe that, and resented being told I did, just made two more people jump on the "Gneech hates gays" bandwagon. Eventually there was nothing to do for it but walk away.)
Of course, if you do think they have a fair point, acknowledge and own up to that. Truth is more important than your pride.