I believe it's a play (or pun) on the word hear.
Hear has quite a few meanings; it can mean to perceive with the ear (e.g., "I hear the taxi blowing its horn"), or it can mean to fully listen to a point of view (e.g., "The judge will hear the case tomorrow").
The Andersen story is a tale about how gossip can result in distortion of the truth.
As she [the hen] settled herself on the perch, she plucked herself with her
beak, and a tiny feather came out.
"There it goes," she said. "No doubt the more I pluck, the more
beautiful I will get." But she said it only in fun, for she was
considered the jolliest among the hens, although, as we've said
before, most respectable. Then she fell asleep.
There was darkness all around, and the hens sat closely together. But
the hen that sat closest to the white hen was not asleep; she had
heard and had not heard, as one should do in this world, if one wishes
to live in peace. But still she couldn't resist telling it to her
"Did you hear what was said? Well, I don't want to mention any names,
but there is a hen here who intends to pluck out all her feathers just
to make herself look well. If I were a rooster, I would despise her."
I think the phrase in question means, "She heard the words, but she did not hear – or perceive – their meaning." In other words, she thought the hen was being vain, when in fact the hen was merely making a lighthearted joke. Thus begins the chain of events that slowly distort the original comment.