Been mulling over why this is, because I've seen takes that it's due to people devaluing filmmaking etc. and that just does not feel right to me (though it may be a knock-on effect).
I think there's two causes. One is pretty simple: we like to feel like we know something more casual fans don't. You rewatch LotR and know that Aragorn's scream is fuelled by a broken toe, and you are an insider in that moment. If you tell other people about this, you get to know that they know you're an insider, and, more altruistically, you get to give them the gift of being insiders as well.
The other is one I maybe don't have the vocabulary to describe. Media studies thing. We are all aware that tv and movies are written, that scenes are filmed in many takes and often out of order, that directors are telling the actors what to do, that wardrobe and makeup workers are making it look like no time has passed or the reverse, that editing may chop it up so that the mood in the finished product is totally different than the mood on set. All of this is done so that you possibly think about it after the fact but not during, because you are supposed to be caught up in the performance. We like the idea that through all of this artificiality, there is a core of something real. Sparks of inspiration that came because an actor felt so in touch with their character that they knew they'd put their hand here instead of pulling back. The dozen people on set controlling every aspect of the scene recognized that magic was happening and gave the couple of actors they're all focused on free rein. A guy flubbed his line and grinned ruefully, breaking character, but it worked as in-character so they left that accident in. (Or he ate that pasta and it made him happy so he shoulder-danced a little.) There is something very compelling to the idea of this magic, amplified when the viewer has a parasocial relationship to the actors involved - or even the director, as the active "we decided to leave that in" decision is a necessary part of the story as well.
And I think the cast and crew are aware of it too! That's why stories about this happening usually come out in press junkets or at cons later on. They know that audiences like hearing about it, and it's also a point of pride for actors to improv something that works and/or highlights how close to the character they became (IME, anyway). I'm sure that there are times when someone goes, hey, no, we made that choice deliberately and it's insulting to portray it as an accident, but I don't think that's the usual response to people talking about improv or being deep in character.