Администратор сказал, что в хостеле нет свободных комнат.

Because Russian past tense verbs agree with the gender and number of the subject.

• администратор is a masculine noun
• so the past tense form is сказал = said for a masculine subject

Compare:

• администратор сказал = the administrator said
• администраторша сказала = the female administrator said
• они сказали = they said

In the past tense, Russian does not change the verb for person the way English does; it mainly changes for gender and number.

Что introduces a subordinate clause and often means that.

So:

• Администратор сказал = The administrator said
• что в хостеле нет свободных комнат = that there are no free rooms in the hostel

This is very common in Russian after verbs of saying, thinking, knowing, and so on:

• Он сказал, что... = He said that...
• Я думаю, что... = I think that...
• Мы знаем, что... = We know that...

In English, that is often optional, but in Russian что is normally kept.

Because в хостеле answers the question where?, not where to?

Russian uses:

• в + accusative for motion into somewhere
• в + prepositional for location inside somewhere

Here the meaning is in the hostel, so it is location:

• в хостеле = in the hostel

Compare:

• Я иду в хостел. = I am going to the hostel.
• Я живу в хостеле. = I live in the hostel.

The noun хостел becomes хостеле in the prepositional case.

Russian commonly uses нет to mean there is no / there are no / not available.

So:

• в хостеле нет свободных комнат = there are no free rooms in the hostel

This is one of the most important patterns in Russian. Instead of saying something like there are not, Russian often says:

• У меня нет времени. = I don’t have time.
• Здесь нет людей. = There are no people here.
• В магазине нет молока. = There is no milk in the store.

Think of нет as a special word meaning there is not / there are not / absent.

Because after нет, the thing that does not exist is usually put in the genitive.

The base form is:

• комната = room

Plural nominative:

• комнаты = rooms

But after нет, Russian uses genitive plural:

• нет комнат = there are no rooms

So in this sentence:

• свободных комнат = of free rooms / any free rooms

This is a very common rule:

• нет времени = no time
• нет денег = no money
• нет мест = no seats / no places
• нет свободных комнат = no free rooms

Adjectives must match the noun they describe in:

• gender
• number
• case

Since комнат is:

• plural
• genitive

the adjective свободный must also become plural genitive:

• свободных комнат

This is just agreement.

Compare:

• свободная комната = a free room
• свободные комнаты = free rooms
• нет свободных комнат = there are no free rooms

Here свободный means available, vacant, or unoccupied, not free of charge.

So свободная комната in a hotel or hostel context means:

• a room that is open
• a room nobody is staying in
• a vacant room

Russian свободный can mean different kinds of free, depending on context:

• Я свободен вечером. = I’m free this evening.
• свободное место = a free seat / available place
• свободная комната = an available room

So context matters.

Russian word order is more flexible than English, but this order is very natural and neutral.

• в хостеле sets the location first: in the hostel
• нет gives the main idea: there is no / there are no
• свободных комнат tells what is lacking: free rooms

So the structure is roughly:

• location + absence + thing absent

Other word orders are possible, but they may sound more emphatic or stylistically different. For a learner, this sentence is a good standard model.

Usually no, not in normal standard Russian with this kind of clause.

After сказал, Russian normally uses что to introduce reported speech or reported information:

• Он сказал, что придёт. = He said that he would come.

Unlike English, where that is often dropped, Russian usually keeps что.

There are other constructions in Russian, but for a sentence like this, что is the expected choice.

Because Russian does not need a dummy subject like English there.

English says:

• There are no free rooms

Russian simply says:

• Нет свободных комнат

Or with location:

• В хостеле нет свободных комнат

Russian expresses existence and absence without a word equivalent to English grammatical there.

Yes. Нет does not change for singular or plural the way English does.

It can mean:

• there is no
• there are no
• not available
• is absent / are absent, depending on context

Examples:

• Нет времени. = There is no time.
• Нет комнат. = There are no rooms.
• Его нет дома. = He is not at home.

So you understand the number from the noun, not from нет itself.

Their dictionary forms are:

• хостел = hostel
• комната = room

In the sentence they change because of case:

• в хостеле

  • dictionary form: хостел
  • case: prepositional singular

• комнат

  • dictionary form: комната
  • case: genitive plural

This is normal in Russian: you often need to work backward from the sentence form to the dictionary form.

Literally it is The administrator said, but in context it may refer to the person working at the desk, so English might translate it as:

• The administrator said
• The receptionist said
• The front-desk clerk said

The exact English choice depends on context. In Russian, администратор is a common word for staff at a hotel, hostel, salon, clinic, and similar places.

Not by itself.

Сказать means to say. If Russian wants to specify the person spoken to, it usually adds it:

• Администратор сказал нам, что... = The administrator told us that...

Without нам or another indirect object, the sentence simply means:

• The administrator said that...

English sometimes uses told more naturally, but grammatically the Russian verb here is just said.