Na de lunch wordt het rustiger in de bibliotheek.

Because when a Dutch sentence starts with something other than the subject, the finite verb usually comes next.

So:

• Na de lunch = first position
• wordt = finite verb, so it must come second
• het = subject-like dummy pronoun

This is a very common Dutch word order pattern:

• Na de lunch wordt het rustiger.
• In de winter is het koud.
• Morgen ga ik naar Amsterdam.

This is often called inversion from an English-speaking learner’s point of view.

Here, wordt comes from the verb worden, which means to become or to get.

So het wordt rustiger means:

• it becomes quieter
• it gets quieter

It is not a passive here. It is describing a change in situation.

Compare:

• Het wordt donker. = It is getting dark.
• Het wordt druk. = It is getting busy.
• Het wordt rustiger. = It is getting quieter.

In this sentence, het does not refer to one specific thing like a book or a room. It is a dummy subject, similar to English it in sentences like:

• It is cold.
• It gets busy here.
• It becomes quieter after lunch.

Dutch often uses het in general statements about weather, time, atmosphere, or a situation.

So het wordt rustiger means something like the situation becomes quieter or simply it gets quieter.

Because rustiger is the comparative form of rustig.

• rustig = quiet / calm
• rustiger = quieter / calmer

The sentence is talking about a change: after lunch, the library is more quiet than before.

This is just like:

• groot → groter = bigger
• druk → drukker = busier
• rustig → rustiger = quieter

In Dutch, short adjectives usually form the comparative with -er, not with meer.

So the normal form is:

• rustiger = quieter

Not:

• meer rustig

You do sometimes use meer with longer or less easily inflected adjectives, but for a common adjective like rustig, rustiger is the standard form.

Because it is used predicatively, not directly before a noun.

Compare:

• de rustige bibliotheek = the quiet library
  
  • here the adjective comes before a noun, so it usually gets -e
• de bibliotheek is rustiger = the library is quieter
  
  • here the adjective comes after a verb, so there is no -e

In your sentence:

• het wordt rustiger

the adjective is part of the predicate, so rustiger stays without -e.

Because lunch is a de-word in Dutch: de lunch.

So you get:

• na de lunch = after lunch

Dutch often uses an article in expressions where English may or may not use one, depending on style. In this sentence, na de lunch is perfectly natural and idiomatic.

Because Dutch often puts place information later in the sentence, especially after the verb and the main complement.

The core of the sentence is:

• Na de lunch wordt het rustiger

Then in de bibliotheek adds where this happens.

That word order sounds very natural in Dutch.

You could move things around for emphasis, but this version is the most neutral:

• Na de lunch wordt het rustiger in de bibliotheek.

Yes. That is also correct, but there is a slight difference in feel.

• wordt het rustiger = it becomes / gets quieter
  
  • emphasizes the change
• is het rustiger = it is quieter
  
  • emphasizes the state

So:

• Na de lunch wordt het rustiger in de bibliotheek.
  suggests that after lunch, the situation changes and it gets quieter.

• Na de lunch is het rustiger in de bibliotheek.
  states the general fact that it is quieter after lunch.

Both are good; wordt highlights the transition more clearly.

Yes. Bibliotheek is a de-word:

• de bibliotheek

So:

• in de bibliotheek = in the library

This is something learners usually just have to memorize with the noun.

Wordt is spelled with dt because of Dutch verb spelling rules.

It comes from worden:

• stem: word
• hij/zij/het form adds -t
• so written form: wordt

Even though it is written dt, the final sound is basically just t in normal pronunciation.

So learners often notice:

• spelling: wordt
• pronunciation: roughly wort

This happens because final voiced consonants are often devoiced in Dutch pronunciation.

Yes. Dutch allows some movement, depending on emphasis.

For example:

• Na de lunch wordt het rustiger in de bibliotheek.
• Het wordt na de lunch rustiger in de bibliotheek.

Both are possible, but the first one is more natural and common if you want to set the time frame first.

Starting with Na de lunch gives the sentence a very natural flow: After lunch, ...

It can be understood either way, depending on context.

It can mean:

• a general truth or routine:
  After lunch, it gets quieter in the library.
• a specific situation today:
  After lunch, it’s getting quieter in the library.

Dutch often uses the present tense for both general statements and near-present situations, just like English sometimes does.

The wider context tells you which meaning is intended.