First of all, there is nothing difficult to understand in the first half of the first sentence, which means the inhabitants of the moon (fairy tales? You can occasionally see a part of our earth in sunny places.

The second half of the key sentence. Because it says that only part can be seen, let's go on to say that the rest is illuminated by the moon. In fact, the second half of the sentence is omitted, and the complete form is

Occasionally, lunar residents will see that part of our earth is completely under the sun, while lunar residents can only see that the rest is illuminated by the light of the moon.

A clause is omitted here. As I said elsewhere, subject+verb ing= subject +that/which/what/who+ verb form. For example.

The boy who asked this question is my roommate, so the omission is that the boy who asked this question is my roommate.

In this way, whoever withdraws and asks will be changed to ing form. This kind of ellipsis is often used by foreigners and is a very important grammar. If it is passive, it is subject+verb participle = subject+what/that/which/who+is/am/are+verb participle. For example.

The bike that was stolen last night is worth 100 dollars. This car was stolen yesterday, worth 100 yuan. Then the situation is omitted.

The bike stolen last night is worth 100 USD.

Here is to get rid of which was and use the verb past tense directly.

Go back to this sentence. The rest is only illuminated by moonlight. This is an omission of a passive cluster sentence, and the complete form is the rest that is only illuminated by moonlight.

So the complete form of the last sentence is

Occasionally, lunar residents will see that a part of our earth is completely under the sun, while lunar residents can only see the rest illuminated by the moon's light.

It means that the new moon can occasionally see part of the earth in full light and other parts in moonlight.

Then the last sentence is easy to understand. They may call it "the old earth in the arms of the new earth". They call it "the old earth on the arms of the new earth". There should be a better translation of this sentence, but it is literally like this. The latter sentence is easy to analyze. The key is that you understand the omission of the last sentence.

This is a more authoritative explanation. Because my English is very professional.