To explain something is to provide a causal account of it, whether in terms of efficient or final cause.
So…
Q: In this context, can the (moon people’s) responses be considered 'explanations'?
A: No. - If they reference their moon god as a cause, however, then it’s an explanation.
Q: Would this vary depending on different philosophical or scientific perspectives?
A: No
Q: Does an explanation need to involve demonstrating how an observation or phenomenon is a consequence of laws, principles, or more fundamental facts?
A: No - Here is an illustration. Q: “Why did you go Grandma’s house? What is the explanation?” A: “I went there because she made cookies and I wanted to eat them. The idea of me eating the cookies is the cause that led to all the behaviors that led up to my arriving at Grandma’s house”. (Final cause) All of this is in Kant, incidentally.
Q: Or can it be simply an account that makes something understandable or clear, in this case, by describing the narrative or beliefs from the book of the Moon?
A: No.
Such is my position.