A problem that can make a story Anvilicious is if the moral in question is common. Even if the moral in question is an important one, people will become desensitized or annoyed with the aesop that the writer is trying to get across if they've heard it often enough, and the message will ring hollow because the story is preaching to deaf ears.
Another issue that can lead to this is Follow the Leader. If a popular work handled an aesop in a nuanced, objective way, then works that come after it might try to preach the same aesop to cash in on the success of that work. These imitators can have a negative effect on the message in question because they can turn it into a trend, and come across as disingenuous. They might not care about the message they're getting across so much as wanting to succeed like the original work they're following.
When people judge a story, they generally care about the plot and characters before the message or theme, and want the latter two to compliment the former two. A good story requires good characters after all. However, an Anvilicious work can be so focused on the message that the characters of the story are either preachy and self-righteous, or Obviously EvilStraw Characters, neither of which are easy for the audience to relate to.