Lütfen içeride bekleyiniz; isimler anons edildikten sonra içeri alacağız.

Both mean please wait.

• bekleyin is the normal polite/plural imperative in everyday Turkish.
• bekleyiniz is a more formal version, often used in announcements, notices, offices, hospitals, and official speech.

So Lütfen içeride bekleyiniz sounds polite and institutional. It can be said to:

• more than one person, or
• one person respectfully.

This is a very common point of confusion.

• içeride = inside, indoors, in the inside area
  It answers where?
• içeri = inside / in, but with the idea of movement toward the inside
  It answers to where?

So:

• İçeride bekleyiniz = Wait inside
• İçeri alacağız = We will take/let you inside

Because the two clauses are talking about two slightly different things:

• içeride bekleyiniz = stay in the waiting area / remain inside
• içeri alacağız = we will admit you into the next room, office, or examination area

In English, both may look like inside, but Turkish distinguishes:

• location: içeride
• direction/movement: içeri

So the sentence is perfectly natural even though both words come from the same root idea.

It means after being announced or more naturally after the names have been announced.

Breakdown:

• anons edil- = to be announced
• -dikten sonra = after doing / after being done

So:

• anons edildikten sonra = after it is/has been announced
• isimler anons edildikten sonra = after the names have been announced

This -dikten sonra structure is extremely common in Turkish:

• geldikten sonra = after coming
• yemek yedikten sonra = after eating
• bittikten sonra = after it finishes

Yes. It is passive.

• anons etmek = to announce
• anons edilmek = to be announced

So isimler anons edildikten sonra literally has the sense of:

• after the names were announced
• after the names have been announced

Turkish often uses passive forms in official or impersonal instructions, especially when the person doing the action is not important.

Because anons is a noun/loanword, and Turkish often turns such words into verbs with etmek.

So:

• anons etmek = to make an announcement / to announce
• passive: anons edilmek = to be announced

This pattern is very common:

• kontrol etmek = to check

• kontrol edilmek = to be checked

• iptal etmek = to cancel
• iptal edilmek = to be cancelled

So anons edildikten sonra follows a very standard pattern.

alacağız is:

• future tense
• 1st person plural
• meaning we will take / we will admit / we will let in

It comes from almak.

Breakdown:

• al- = take
• -acak/-ecek = future
• -ız/-iz/-uz/-üz = we

So alacağız = we will take

In this sentence, the natural meaning is not a literal physical taking, but we will call/admit you in.

It is omitted, because Turkish often leaves pronouns out when they are understood from context.

So içeri alacağız literally looks like we will take inside, but the object is understood as you.

A fuller version could be:

• Sizi içeri alacağız = We will let you in

Turkish commonly drops sizi when it is obvious who is being talked to.

Literally, yes:

• içeri almak = to take inside

But in many everyday situations it is an idiomatic expression meaning:

• to let in
• to admit
• to call someone into the room/office

For example, at a clinic, office, or interview room, Sizi içeri alacağız usually means:

• We’ll let you in shortly
• We’ll call you in

So this is a very useful expression to learn as a set phrase.

Because isimler is the subject of the passive clause, not a direct object.

In isimler anons edildikten sonra:

• isimler = the names / names
• they are the thing that gets announced

Since the verb is passive, isimler functions like the subject of are announced, so it stays in the basic form.

Also, Turkish does not have a word exactly like the, so isimler can mean:

• names, or
• the names

The context tells you it means the names here.

It is fairly formal and polite.

That feeling comes from several things:

• Lütfen = please
• bekleyiniz = formal imperative
• passive wording like anons edildikten sonra
• future alacağız, which sounds orderly and official

This is the kind of sentence you might hear or read in:

• a hospital
• a government office
• a school office
• an interview area
• a waiting room

A more everyday version might be something like:

• Lütfen içeride bekleyin; isimler anons edildikten sonra sizi içeri alacağız.

That says basically the same thing, but with a slightly more common spoken style.