In out of the dust why does Billie Jo clean up the mud even though it hurts her hands?

Answer 1

She doesn't want the mud to get her hands infected.

Answer 2

Billie Jo cleans up the mud because she is trying to cope with the grief and guilt she feels after the accident that killed her mother and left her injured. It is a way for her to distract herself from the pain and to try to regain a sense of control over her life. Cleaning up the mud is a physical manifestation of her internal struggle to make things right.