During the end of the Doll House play, the sentence says "from below, the sound of a door slamming shut." What does this mean?

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.