I get that the question means "Why is the hotel full?" but I don't really get why "il" is in there, since it seems like the subject is already established.
If I wanted to ask, "Why is it full?" , do I say "pourquoi il est" eller "pourquoi est-il?" and why?
Even though this is more or less a duplicate, I'll add an explanation that I didn't see when skimming previous answers.
The subject pronouns are clitics, which have some fascinating properties but are perhaps best summarized as being between words and affixes. They're smaller and less independent than words, but more than affixes.
Now let's see how this plays into the situation you cited.
In the normal question structure, the noun comes before the verb:
L'hôtel est complet.
In the inverted question structure, it gets switched around:
Est l'hôtel complet ? (not correct yet)
We need something in that spot after the verb to tell us what the subject is. However, in French syntax, a noun slot after a verb isn't quite as strong as a slot before a verb. See this resource top of p. 142: "While a preverbal subject is usually topical, a postverbal subject is never topical." The term "topical" carries various meanings, but what matters is that important information like the topic of the sentence can't go in this postverbal slot.
This means that we can't "fit" l'hôtel there. So we move it back before the verb, in the stronger spot at the beginning of the sentence:
L'hôtel est __ complet ? (not correct yet)
The reason I leave the __ there is because when you move things in syntax, you leave a gap that often needs to be filled by something. "Est" still wants to be followed by the subject of the verb.
So what can fill that gap? Something smaller: a clitic pronoun. It will "resume" or double the subject, but be weak enough to fit into the postverbal slot. It's not topical in itself, but a reference back to the topic.
That gives us:
L'hôtel est-il complet ?
That paper I linked to is technical, but a skim of pp. 141-8 could yield more insight.
To summarize, the rules apply like this:
For your question about "Why is it full?", you have two options. If you've already introduced the topic, so it's clear you're talking about the hotel:
Pourquoi est-il complet ?
But if you still have to introduce the topic l'hôtel, it can only go before the verb:
Pourquoi l'hôtel est-il complet ?
Don't forget that there is also an informal question structure that isn't inverted at all. That's why you'll also see:
L'hôtel est complet ?
This is just a normal sentence structure, with intonation used to mark the question instead of inversion. This can be heard in conversation, but it's not suitable for more formal contexts.
While we're pausing on subject pronouns, note that another consequence of their being clitics and not full words means that they're unable to provide the kind of stress that English speakers tend to demand of them. An example would be when you want to insist on this person and not that person.
Sometimes you can solve this issue by mentioning the noun or the person's name again. But when there really is no reason to repeat yourself, you can use the disjunctive pronoun instead -- and sound very French doing so. :)
Consider this quote from the Bible (John 3:30):
"He must increase, but I must decrease."
An English speaker who wanted to stress that second pronoun might be tempted to italicize it:
"He must increase, but I must decrease."
Now, here's the French translation:
"Il faut qu'il croisse et que je diminue." (Louis Segond)
If you try to do the same emphasis in French, you get a problem:
"Il faut qu'il croisse et que je diminue."
This sound odd to French ears, because the clitic is not "strong" enough to be stressed. Instead, the disjunctive pronoun appears:
"Il faut qu'il croisse et que moi, je diminue." (Louis Segond 21)
I think it's a phenomenon that happens in some languages for "emphasis". For example, Albanian: i-a dhashë librin atij, equivalent to English "I gave the book to him", but translated literally as "him-it I-gave the-book to-him". It's also reminiscent of Romanian: Scrisoare lui Neacșu, where the personal pronoun lui could be another subject to scrisoare (the letter) but mainly serves as a "fracta" genitive case used for proper names, whereas such a phrase as "the name of the rose" would be rendered as numele triandafirului, triandafirul with the vestige of the Latin personal pronoun ille at the end being the nominative for "rose".
