Magister dicit discipulos in bibliotheca tacere oportere.

Because this sentence uses indirect statement after dicit.

After verbs like say, think, know, and hear, Latin often uses:

• accusative subject
• • infinitive verb

So in discipulos in bibliotheca tacere oportere, discipulos is the subject of the reported statement, but Latin puts that subject in the accusative.

So:

• magister = the teacher, subject of dicit
• discipulos = the students, subject of tacere oportere inside the reported clause

The whole reported idea is what the teacher says.

Because Latin usually does not use a separate word like English that in this kind of sentence.

English says:

• The teacher says that the students ought to be silent.

Latin normally says:

• The teacher says [students to ought to be silent].

That sounds strange in English, but it is the normal Latin pattern. This is called the accusative-and-infinitive construction.

They do different jobs.

• oportere is infinitive because it is part of the indirect statement after dicit
• tacere is infinitive because it depends on oportere

So the structure is basically:

• dicit = says
• discipulos ... oportere = that the students ought
• tacere = to be silent

In other words, oportere takes tacere as its complement:

• oportere tacere = to ought to be silent / to have to be silent

A natural English rendering is simply that the students ought to be silent.

Oportere is the infinitive of oportet, an impersonal verb meaning something like:

• it is proper
• it is necessary
• it is right
• one ought

In Latin, oportet often appears with:

• a person in the accusative
• plus an infinitive

So:

• oportet discipulos tacere = the students ought to be silent

In your sentence, that whole idea is put into indirect statement after dicit, so oportet becomes oportere:

• Magister dicit discipulos tacere oportere.

Tacere is the present active infinitive of taceo.

Its basic meaning is:

• to be silent
• to keep quiet
• to say nothing

So here it means that the students ought to keep quiet.

Because it is a present infinitive, it normally shows action happening at the same time as the main verb of saying:

• The teacher says ... the students ought to be silent.

Because in changes meaning depending on the case.

• in + ablative = in / at a place
• in + accusative = into a place

So:

• in bibliotheca = in the library
• in bibliothecam = into the library

Here the sentence is talking about location, not movement, so the ablative bibliotheca is correct.

Here is the breakdown:

• magister is the subject of dicit
• discipulos is the logical subject of tacere
• oportere is impersonal, so it does not have a normal nominative subject like an ordinary personal verb

So the teacher is the one saying, but the students are the ones who ought to be silent.

A very literal unpacking would be something like:

• The teacher says the students in the library to be silent ought.

That is not good English, but it shows the Latin structure.

A smoother English understanding is:

• The teacher says that the students ought to be silent in the library.

Latin word order is much more flexible than English word order, so learners should focus more on:

• case endings
• verb forms
• constructions

than on translating word by word in order.

A simple main-clause version would be:

• Oportet discipulos in bibliotheca tacere.

That means:

• The students ought to be silent in the library.

Then, after dicit, Latin changes oportet to the infinitive oportere as part of indirect statement:

• Magister dicit discipulos in bibliotheca tacere oportere.

Seeing that pair side by side often makes the sentence much easier to understand.