What is the significance of the "Little Shirley Beans" record and the fact that Holden breaks it?

The record, "Little Shirley Beans", is symbolic of childhood and the sense of innocence of which Holden is afraid to let go.  Fittingly, the song is "about a little kid that wouldn't go out of the house because two of her front teeth were out and she was ashamed to".  Like Holden, the little girl in the song is afraid of the changes that accompany growing older, in her case, the loss of her baby teeth. Holden pays five dollars for the record, which is a lot of money. Symbolically, he would pay dearly to be able to avoid the reality of growing up (Chapter 16). 

When the record breaks "into about fifty pieces", it represents the dissolution of Holden's dream of being able to avoid entering the world of adulthood.  Holden had tried to protect his dream of childhood and innocence, keeping it carefully "in a big envelope and all", but despite his best efforts, the record still shatters.  This indicates that, try as he might, Holden is going to be forced to let go of the past and face the future, leaving the security of childhood behind (Chapter 20).