We're never really told what is wrong with the woman. However, she tells us that she has a baby. She also says that her husband rented the house for her to rest because of her "nervous condition." So we can infer that she is suffering from postpartum depression.
Think of what you might find in a nursery. A baby crib has bars, and a playpen has some other type of restraint. The windows of her room are barred, she thinks to protect the children--but really to keep her in? Just as parents do to protect small children, someone has placed a "gate" at the head of the stairs, preventing her from leaving her designated area. Parents use pet names for their children; I'm always calling my niece a silly goose. The woman's husband calls her a "blessed little goose." Her bed is even nailed to the floor!
The woman, who is imbalanced (maybe a touch unbalanced) after the birth of her baby, has herself become like a baby. Just as a nursery is designed to protect the infant, so also her room has been designed to confine her. Whereas a nursery is usually associated with joy, her "nursery" is a place of confusion, sadness, and pain.
There are multiple reasons why this is significant. One relates to the plot: it gives a reason why the furniture in the room might be the way it is. It's an excuse. More, though, it is important for purposes of social critique. That is to say, it shows the relationship between men and women at that time. The husband is essentially treating the wife like she's an infant. This would give him some reason to reject or dismiss her comments too; even before she's crazy or trapped in the wallpaper, any comment that doesn't make sense is just her acting out. It's baby talk.