Maybe you want to let people give opinions without fear of judgment. Maybe you just want to allow silly confessions. As for me, back in the old days of livejournal, there was a whole subculture of submitting anonymous story prompts using the anon reply feature and other people writing those prompt fills (also anonymously) for fun. It died out as livejournal died, but it would be a fun medium to revive.
Assuming this is something that could be toggled by the poster on a per-post basis, I think this sounds like it could be a lot of fun for smaller accounts. I definitely worry about it being a vector for harassment, however, and I could see it being unusable for anyone with a following over a certain size.
For this to be implemented, I think it would need to be something that could also be turned off after a post was created, and a big switch to disable anonymous comments on all past posts would also be useful. (I'm envisioning a scenario where a user becomes a target of harassment, and the harassing users go back through their page to find old posts with anonymous comments turned on.) Additionally, the ability to fully delete anonymous comments under your posts would be good.
Yesss this would be so fun, I love this idea and the suggestions from jgs and Bazelgeuse-apologist.