Ready for some crumbs of wisdom? As with the red shoes, a loaf of bread can symbolize both sinfulness and redemption.
Inger, from "The Girl Who Stepped on Bread," is a major brat. Not only does she pull the wings off flies and impale other insects, she also refuses to go visit her poor parents once she gets a swank servant-girl position. When her new mistress finally makes her go visit her folks, Inger's given a "big loaf of white bread" (91.11) to bring along.
Why does this even matter? Hello, her parents are rural poor people in 19th-century Denmark. Of course having free food dropped into your lap matters. But Inger's so obsessed with looking good that "when finally a big puddle blocked her way, she threw the bread into it rather than get her shoes wet." (91.12) Big mistake, since she sinks down into the bog and becomes frozen as a statue in hell for, like, decades.
Disrespecting bread—a.k.a. disrespecting your family and depriving them of food—is therefore a big sin. It's totes selfish.
When Inger realizes how awful a sinner she is, she's transformed from a statue into a sparrow. At first she just flies around and tries not to draw too much attention to herself, but then she starts to collect bread to give to the other hungry birds during the winter: "At the places where travelers had rested, she would sometimes find little pieces of bread and crumbs. She ate very little herself but called the other starving sparrows, so that they could eat" (91.50).
In this way, the newly selfless Inger-bird begins to redistribute bread to the needy. Eventually, she manages to give away "so many crumbs that they weighed as much as the bread that Inger had stepped on, in order not to dirty her shoes" (91.51). And a lot fewer hungry birds isn't the only measure of success for Inger-bird's Bread Redistribution Program; her Robin Hood-ing seems to grant her a ticket to heaven.
Being willing to give up your bread/food for others is a great way to show that you're not only into yourself, and your own well-being. This alone may or may not be enough to get you into heaven, but for Inger, it did the trick—in combination with realizing how awful a sinner she was, and repenting within her heart. But overall, in Andersen's tales, bread is a measure of one's care for others. Put that in your oven and cook it.