I looked up blithe in the dictionary, which means "happy and without worry." For example, "She shows a blithe disregard for danger." I just don't understand what a blithe disregard means in the context.
In this context, you use "blithe" when something that would normally cause great concern is not a concern to the subject. The subject does not care about something that people would normally care about. This lack of caring frees the subject to do something unusual. This lack of caring allows the person to be "happy and without worry" when a normal person would worry and therefore not be happy.
In "she shows a blithe disregard for danger", we would expect someone to show concern for danger (avoiding it, fearing it, approaching it carefully, etc.), but she defies that expectation. It is as if she is immune to whatever concerns would normally hold someone back from danger. Maybe she does not care about her own safety. Maybe she thinks she is invincible, or maybe she just does not care if she gets hurt.
The adjective blithe used to mean happy and carefree, but over time it has also come to describe someone who isn't paying attention the way they should. (vocabulary.com)
carefree and happy and lighthearted
“was loved for her blithe spirit”
lacking or showing a lack of due concern
“spoke with blithe ignorance of the true situation”
To be blithe means to be unconcerned and carefree, especially in situations where one should be heedful of risk to oneself or to others.
The chemical manufacturer showed a blithe disregard for the health and safety of the town's residents when it discharged toxic waste into the stream.
The explorer blithely crossed the crocodile-infested river in a small dugout canoe. He smoked his meerschaum pipe while the guide paddled.