Many pianists are raised (as I was) with a lot of hifalutin vocabulary about "meaning" in music. The more seriously I pay attention to the way children experience sound the more I believe that the meaning--what one student called "the music part" is inherent in the act of perception.
It is extremely specific. Take a sound that is stable--the perfect fifth A E, for example. Changing the A to D a 9th below E, is enough to make a child cry or want to stop playing the piece. In effect the child is expressing a question: What have you done to my beautiful E? In time the child can learn to pose subsequent questions, such as: Will my beloved E ever come back? Why would anyone do such a thing on purpose?
Every such question, with or without an answer, yields the ability to play the piece as if telling a story--every time.