There isn’t a single simple answer.
I find people who have been abused, or perceive themselves as abused, have a very low threshold for emotional discomfort. It’s like a PTSD of sorts. This includes mental abuse. If they always felt like they were being verbally beaten down or dismissed, they are trying to come to their own defense before you (or whoever it is) does more damage. It’s a self defense.
I would also say insecure people can be easily offended.
Fearful people tend to be easily offended.
People with trust issues tend to be easily offended.
When it’s fsmily sometimes it’s just a dynamic that has developed charged with emotion. It’s like when I would babysit my niece and nephew they behaved. Not because I was fantastic, but because they were rarely with me. On the other hand with their mom, they constantly challenged her and caused some havoc. They might be able to listen to advice and ideas from me with less emotion than listening to their mom. It’s just the position in the family.
Lastly, sometimes it’s people who they themselves are judgemental are quick to feel judged. They are projecting. Like when the cheating husband doesn’t trust his wife. Well yeah, he believes people cheat, because he is a cheater.
There is a Q right now about regarding how we think, and how it affects our lives. Most people on that Q agree that how we control our thoughts and view the world really matters. I think this is also entangled with taking offense.
Sometimes, the person is trying to help the offender. They aren’t so much taking offense, but telling the offender they are screwing up. Like when a student from U. Mich who was visiting at State said to her boyfriend after I said hello to them while waiting for an elevator, “people are so nice here, people at UM aren’t friendly they’re all a bunch of Jews,” and I told her the what for in a very nice way, I wasn’t offended, I just think she’s an idiot. She was being offensive, and if that isn’t her intent she needs to rethink and learn she sounds like an idiot and a bigot.