Learn Hardcore Croatian: Djeca šapuću u knjižnici kako ne bi smetala odraslima. - The children are whispering in the library so that they don’t disturb the adults.

kako here means “so that / in order that” (expressing purpose).
ne bi is the conditional auxiliary (“would not”).
smetala is a past active participle agreeing in gender/number with djeca (neuter plural → ending -la).

So the whole clause is a purpose clause:

Djeca šapuću … kako ne bi smetala odraslima.
“The children whisper … so that they wouldn’t disturb the adults.”

Could you use da instead of kako? For example: … da ne bi smetala odraslima?

Yes, you can say:

Djeca šapuću u knjižnici da ne bi smetala odraslima.

Both kako and da are used to introduce purpose clauses.
Nuance:

kako ne bi… is slightly more formal / stylistic in written language.
da ne bi… is very common in speech.

Meaning is practically the same: “so that they wouldn’t disturb… / in order not to disturb…”.

Why is it ne bi smetala and not just ne smetaju?

There’s a difference in meaning:

ne smetaju odraslima = “they do not disturb the adults” (a simple fact).
kako ne bi smetala odraslima = “so that they would not disturb the adults” – this expresses intention/purpose and a possible action that is being prevented.

The conditional (bi + participle) is used to show:

a potential action (they might disturb)
which is being avoided by whispering.

So the structure kako ne bi + participle is standard for “so that [something] would not happen”.

What exactly is smetala grammatically, and why does it end in -la?

Smetala is:

the neuter plural form
of the past active participle of smetati (“to disturb”).

Past active participle endings:

masculine plural: -li (onI su smetali)
feminine plural: -le (one su smetale)
neuter plural: -la (ona su smetala)

Since djeca is grammatically neuter plural, the matching participle is smetala:

(djeca) bi smetala
(djeca) su smetala

What case is odraslima, and why is that case used with smetati?

Odraslima is dative plural of odrasli (“adults”).

Pattern:

nominative plural: odrasli – “adults”
dative plural: odraslima – “to (the) adults”

The verb smetati (“to bother/disturb”) takes an indirect object in the dative:

Smetati komu? – “to disturb whom?” (dative)
Smetam susjedima. – “I disturb the neighbours.”
Djeca bi smetala odraslima. – “The children would disturb the adults.”

So: odraslima = “to the adults”.

Why is odrasli used for “adults”? Isn’t that an adjective?

Yes, odrasli is originally an adjective meaning “grown-up”.

In Croatian, adjectives can be nounified (used as nouns) by dropping the noun they describe when it’s clear from context. So:

odrasla osoba = “an adult person”
odrasli (literally: “the grown(-up) ones”) = “adults”

Here, odrasli / odraslima functions as a noun: “adults” / “to the adults”.

Could the word order in the sentence be changed? For example: U knjižnici djeca šapuću kako ne bi smetala odraslima.

Yes, Croatian word order is relatively flexible. All of these are grammatical:

Djeca šapuću u knjižnici kako ne bi smetala odraslima.
U knjižnici djeca šapuću kako ne bi smetala odraslima.

Differences are mainly in emphasis:

Starting with Djeca… emphasizes the children.
Starting with U knjižnici… emphasizes the place first: “In the library, the children are whispering…”

The core grammar (cases, verb forms) stays the same regardless of the order.

What is the aspect of šaptati and smetati, and is that important here?

Both šaptati (“to whisper”) and smetati (“to disturb”) are imperfective verbs.

Imperfective aspect:

describes ongoing, repeated, or habitual actions
is typically used for present tense and for background actions.

In this sentence:

šapuću – ongoing whispering
(ne bi) smetala – a possible ongoing / repeated disturbance

So the imperfective aspect fits well with the idea of continuous behaviour in the library, not a single completed event.

Is there a difference between knjižnica and biblioteka?

Both can mean “library”, but there are usage differences:

knjižnica – preferred in standard Croatian (especially in Croatia).
biblioteka – more common in Serbian, and also understood in Croatian, but in Croatian it can also mean “collection of books / series of books” rather than the institution.

In a Croatian textbook or exam, knjižnica is usually the recommended word for “library”.

Can the sentence also be translated as “The children are whispering in the library in order not to disturb the adults”? Does kako ne bi always mean “in order not to”?

Yes, that is a good natural translation.

In this kind of context, kako ne bi does indeed mean “so that (something) would not / in order not to”. It’s a very common way to express preventing an undesired result:

Zatvorio je prozor kako ne bi bilo propuha.
“He closed the window so that there wouldn’t be a draft.”

So in your sentence, kako ne bi smetala odraslima = “in order not to disturb the adults.”