La biblioteca es igual de tranquila que mi cuarto por la noche.

It means as ... as for a comparison of equality.

So:

• igual de tranquila que = as quiet as

The full pattern is:

• igual de + adjective/adverb + que

Examples:

• Es igual de fácil que el otro. = It’s as easy as the other one.
• Corre igual de rápido que yo. = He/She runs as fast as I do.

In your sentence, tranquila is the adjective being compared.

Because tranquila has to agree with la biblioteca, which is feminine singular.

Agreement in Spanish:

• la biblioteca tranquila
• el cuarto tranquilo
• las bibliotecas tranquilas
• los cuartos tranquilos

Even though English says quiet, Spanish adjectives usually change form to match the noun’s gender and number.

Yes. That is very natural too.

So you could say:

• La biblioteca es tan tranquila como mi cuarto por la noche.

Both mean basically the same thing:

• igual de ... que
• tan ... como

For learners, it helps to remember:

• as quiet as = tan tranquila como
• also possible: igual de tranquila que

Both are correct and common.

Because this comparison pattern requires it before an adjective or adverb:

• igual de + adjective + que
• igual de + adverb + que

So:

• igual de tranquila que
• igual de rápido que
• igual de interesante que

You cannot normally say igual tranquila que in standard Spanish for this meaning.

Here, es presents quiet as a characteristic or general quality in the comparison.

• La biblioteca es tranquila. = The library is quiet / a quiet place.
• La biblioteca está tranquila. = The library is quiet right now.

So:

• es = more general, descriptive
• está = more temporary, situational

In this sentence, the idea is that the library has the same level of quietness as your room at night, so es works well.

Here mi cuarto means my room, often specifically my bedroom.

Cuarto literally means room, but in everyday speech it often means someone’s bedroom when the context is clear.

Other common words are:

• habitación = room / bedroom
• dormitorio = bedroom
• cuarto = room, often bedroom in context

So mi cuarto is a very natural way to say my room.

Por la noche means at night or in the nighttime.

It is part of a very common set of time expressions:

• por la mañana = in the morning
• por la tarde = in the afternoon / evening
• por la noche = at night

You usually just learn these as fixed expressions.

Compared with other options:

• de noche = at night, nighttime, often more general
• en la noche = less typical in Spain for this kind of sentence

So for Spain Spanish, por la noche is a very normal choice.

In this sentence, it most naturally describes mi cuarto, because it comes right after it:

• ... que mi cuarto por la noche

So the most natural reading is:

• The library is as quiet as my room at night.

If you wanted por la noche to clearly describe the library, you could move it:

• Por la noche, la biblioteca es igual de tranquila que mi cuarto.
• La biblioteca, por la noche, es igual de tranquila que mi cuarto.

Word order can help show what the time phrase belongs to.

Because mi already specifies the noun.

In Spanish:

• la biblioteca = the library
• mi cuarto = my room

You do not normally combine a regular definite article with a possessive like that in modern standard Spanish, so not:

• la mi cuarto

The article la is used because biblioteca is being referred to as a specific noun, while mi already does that job for cuarto.

It can mean more than one thing depending on context.

For a place, tranquilo/tranquila can mean:

• quiet
• calm
• peaceful

So this sentence suggests the library is very calm and quiet, not necessarily completely silent.

If you wanted to stress pure silence, you might use:

• silenciosa = silent / very quiet

So:

• tranquila = peaceful, calm, quiet atmosphere
• silenciosa = emphasis on lack of sound

Yes, it is natural and understandable.

A Spaniard might also say:

• La biblioteca es tan tranquila como mi cuarto por la noche.
• La biblioteca es igual de silenciosa que mi cuarto por la noche.
• La biblioteca es tan silenciosa como mi habitación por la noche.

Small vocabulary choices may vary:

• cuarto and habitación are both possible
• tranquila and silenciosa are both possible, depending on nuance

But your original sentence is perfectly good Spanish.