Why does Lyddie think she has been cast off like dry husks to wind?

The quote you mentioned comes from Chapter 14.

In that chapter, Betsy develops a cough and never quite recovers her strength. She becomes so weak that she has to leave her factory job. Betsy's uncle comes to take her back to Maine in April. Before she leaves, Betsy assures Lyddie that she will be back in a year; she also voices her desire to have enough money to attend college one day.

As she sees...

The quote you mentioned comes from Chapter 14.

In that chapter, Betsy develops a cough and never quite recovers her strength. She becomes so weak that she has to leave her factory job. Betsy's uncle comes to take her back to Maine in April. Before she leaves, Betsy assures Lyddie that she will be back in a year; she also voices her desire to have enough money to attend college one day.

As she sees her friend off, Lyddie becomes convinced that Betsy will never be strong enough to work thirteen to fourteen hours a day in the factory again. Lyddie decides that, when she is ready to the leave the factory, she will sign the petition for Betsy's sake. 

For her part, Lyddie feels that it is wrong for the factory owners "to suck the strength of their youth, then cast them off like dry husks to the wind." To Lyddie, the factory owners care only about their profit margin. They make their employees toil for long hours in hazardous working conditions. Young girls like Rachel and Betsy are considered expendable. When workers become sick from the toxic air, they are cast aside. The factory owners take no responsibility for the girls' suffering, which means that the girls have no recourse to competent medical care. In Lyddie's mind, these girls are cast away like "dry husks to the wind." They are treated as no more than encumbrances to be discarded once their health fails.