You’re right, we usually say “the sky”. If you look up, you’ll only see one sky. Here however, we’re talking about the sky over many parts of the world — the sky over Asia, over Africa, over the Atlantic Ocean, etc., so we can say the skies. It give the sense of flying over the whole world, not just in a local area. Words like “land(s)” and “sea(s)” work the same way.

This plural expression is just idiomatic. I agree that we don't normally use the plural form of the word "sky." Just add this expression to your growing list of things that don't make sense in English. I am sure Thai has a similar list. We have a similar expression for ships on the "high seas" (although technically there is more than one sea.)