We're just going to let this quote stand in for the rest of the "example stories" that make up the first part of the poem. Because when you look at them, they're mostly the same: something improbable happens and someone goes "from toilets" (or something similarly icky) "to riches." In each case, though, the luck is based around money—making it seem like the only way to be fortunate is to become rich.
Her father brought presents home from town, jewels and gowns for the other women but the twig of a tree for Cinderella. (33-35)
Here again, the fortune or misfortune of the situation centers on money (or expensive stuff). We think, oh man, that stinks for Cinderella because she didn't get any of the costly things her stepsisters did. Once again, it seems like the only thing that matters in this poem is money.
Next came the ball, as you all know. It was a marriage market. The prince was looking for a wife. (41-43)
Well, the prince might have been looking for a wife for other reasons, but most of the women at that "marriage market" ball are looking for one thing: money. Marrying into a royal family is a surefire way to turn your luck around when it comes to wealth, and that's what every woman in this story seems to want. We don't know much about the prince— maybe he's a total jerk—but it doesn't matter. Marrying him and moving up in society are the only goals here, it seems.
Mama! Mama! My turtledove, send me to the prince's ball! The bird dropped down a golden dress and delicate little gold slippers. (58-61)
Cinderella isn't fit to go to the ball, and why not? Because she's poor. She must look wealthy in order to be noticed by the prince, so she has to wish for things that will make her look wealthy. A gold dress and shoes do the trick. Later the poem says that her stepfamily didn't even recognize Cinderella without her dirty face, which is another way of saying that wealth turns you into a different person altogether.
At the wedding ceremony the two sisters came to curry favor and the white dove pecked their eyes out. (95-97)
The stepsisters here are still trying to reap some of the benefits that come from knowing a royal family, and based on the rest of the poem, those benefits are almost surely monetary. The dove is having none of it: ultimate justice, in this poem, is measured by allowing or denying people wealth. Even though they come from a wealthy family, the stepsisters want more, and are punished for it.
never bothered by diapers or dust (3)
This quote toward the end of the poem brings back all the example stories at the beginning. The "diapers" and "dust" are reminiscent of the duties of the nursemaid and charwoman, and more generally of the things wealthy people don't have to deal with. Saying that Cinderella and the prince are "never bothered" by these dirty things is, in part, saying that they were very rich.