This is a very good question because the statement that you are asking about is the last stage direction we see in the play A Doll's House, by Henrik Ibsen.
There has been somewhat of an argument on what the door shutting implies, but there are several assumptions:
It symbolizes finality: Nora walks off, shutting the door of her home, her doll's house, behind her for good.
It symbolizes change: By leaving the house, Nora is showing the world what she is really made of. She is not afraid. She is not ashamed. She is not even worried. She changed so much that she cannot see herself inside that place. When she shuts the door behind her, she is ready to move on to another life. Torvald, however, stays inside the house unchanged, shocked, and alone.
It symbolizes freedom: Once the door is shut, Nora is gone. Inside, Torvald continues to call her name in disbelief. How could have Nora abandoned him and the children? What happens now? It does not matter much: Nora has been finally liberated from her secret, from an oppressive life, from a fake reality, and from a marriage in which she was nothing but another piece of entertainment.