"I would like to fly you over here" is not rude. It's a pretty neutral sentence. Here, it means that the person saying the sentence will pay for the listener's airplane tickets. :)

P.S. Your two example sentences in the end are the same. Is it supposed to be like that?

Hi, it's interesting. I'd like to see native people's opinion.

あなたをここに飛行機で運んできたい。と言う意味かなあ？

It's basically the difference between what comes first after the word "like."

If you say:

"I would like to clean my room."

In the above sentence, [to clean my room] is an infinitive phrase serving as a noun, which is simply the direct object of the transitive verb [to like].

But, if you say:

"I would like you to clean my room."

Now, the infinitive phrase [to clean my room] is still the direct object of the verb [to like]. [my room] is still the direct object of the action of the infinitive (clean what? clean my room). What has changed is that the word [you] is now the indirect object of the action in the infinite (room cleaned by whom? by you).

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So, looking at the flying example:

"We'd like to fly you over here."

In the above sentence, [to fly you over here] is an infinitive phrase acting as a single noun object of the verb [to like], so you should think of it as a single action, performed by the subject [We]. [you] is the direct object of the infinitive verb (fly who? fly you).

"We'd like you to fly over here."

Now, [you] is outside (before) the infinitive phrase, so it becomes the indirect object of the infinitive verb. In other words, by making [you] the indirect object, you are now saying that the subject [We] would like the indirect object to perform the action of the infinitive - as opposed to the previous sentence, where [you] was wrapped up inside the infinite phrase, and there was no indirect object.

I found a sentence in an English study book, which I really don't understand the usage of the word "fly."
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I hate to be nit-picky, but your sentence above kind of "clanks" in my mind. I'm having trouble with ", which" because I don't know if "which" relates to "book" or "sentence." Well, I mean, I know what you meant, but I think the statement would be clearer if we wrote:

"In an English study book I was reading, I came across [found] a sentence THAT I really don't understand, involving the use of the word 'fly.'" (OR you could just as easily leave out "that." But I added it to explain a point.)

"Comma which" sentences and "no-comma that" sentences are even difficult for most Americans, so don't feel alone! ;-) "Which" describes, while "that" defines (narrrows it down to one).

"This is the car that Mirai bought." vs. "This is Mirai's car, which won car-of-the-year honors recently." The ", which" phrase is not really necessary. It's additional information, whereas "the car that Mirai bought" narrows it down, defines, and the "that" phrase is totally necessary to make your point. I hope this helps!

;-)

"That sounds like the ideal move." -- Is that really in the conversation? It sounds unnatural to me.

これは「I'll drive you home.」の「drive」と似たような感覚でとらえればいいと思います。「drive you」が「あなたを転がす」のではなく「車で運んであげる」という意味になるのと同じように、「fly」の直訳も「あなたを飛ばしたい」ではなく、「あなたを飛行機で運んであげたい」が近いのではないでしょうか。そして、車の場合は「自分で運転して乗せて行ってあげる」という状況が自然に想起できるのに対し、飛行機で運んであげるというときに「操縦して乗せていく」というシチュエーションはなかなか起こりづらいため、言外の含みとして「料金を持つ」という発想になりやすいのだと思います。状況次第では、「自家用ジェットに乗せてあげたい」とか、もっと極端な場合だと「自分が空を飛んで背中に乗せていってあげたい」なんていうときにも、この「fly」は使えるはずです。

As others have said, "We'd like to fly you over here" implies that the speaker (or the speaker's company) will be paying for the tickets.

だから、聞き手が喜ぶでしょう　^^