Moja sestra je htjela staviti novu ogrlicu, ali ju je mačka skinula čim smo došli kući.

Because moj behaves like an adjective and must agree with the noun it describes.

• sestra is feminine singular
• so the correct form is moja
• compare:
  • moj brat = my brother
  • moja sestra = my sister
  • moje dijete = my child

So moja matches the gender, number, and case of sestra.

Because the subject is sestra, which is feminine singular.

In Croatian past-tense forms, the participle agrees with the subject:

• on je htio = he wanted
• ona je htjela = she wanted

So moja sestra je htjela uses the feminine form htjela.

This is the normal Croatian past tense, often called the perfect.

It is built with:

• the present tense of biti = to be
• plus the past participle

So here you have:

• je htjela = she wanted
• je skinula = it/she took off
• smo došli = we came

A very important point: the participle agrees with the subject in gender and number.

• sestra je htjela
• mačka je skinula
• mi smo došli / mi smo došle

Also, the auxiliary words je and smo are clitics, so they usually appear near the beginning of the clause.

Because ogrlicu is the direct object of staviti, so it goes in the accusative case.

The dictionary form is:

• nova ogrlica = a new necklace

But after a verb like staviti = to put on / put, you need the accusative:

• staviti novu ogrlicu

Both words change because the adjective must agree with the noun:

• nova → novu
• ogrlica → ogrlicu

Because Croatian often uses staviti with clothes or accessories where English would say put on.

So:

• staviti ogrlicu = to put on a necklace
• staviti kaput = to put on a coat

The verb staviti basically means put / place, but with clothing and accessories it often naturally means put on.

Likewise:

• skinuti ogrlicu = take off a necklace

So Croatian does not always need a separate particle like English on or off.

Ju refers to ogrlicu.

So in English it means it:

• ali ju je mačka skinula = but the cat took it off

Why ju and not je?

Because the normal unstressed accusative feminine form is often je, but here the next word is also je (the auxiliary verb). Croatian avoids the awkward sequence je je, so it uses ju instead:

• natural: ju je
• not used here: je je

So ju is chosen mainly to avoid confusion and bad sound repetition.

Because Croatian clitics tend to stand very early in the clause, usually in the second position.

Here, ju and je are both clitics, so they are pulled toward the front:

• ali ju je mačka skinula

That is why the object pronoun appears before the full subject mačka.

You can also hear:

• ali mačka ju je skinula

That version is possible too, but it gives a slightly different rhythm or emphasis. The sentence you were given follows a very common clitic-placement pattern.

Because mačka is a feminine noun grammatically, so the past participle must agree with it:

• mačka je skinula

If the sentence used mačak = tomcat / male cat, then you would get:

• mačak je skinuo

So the verb form follows the grammatical gender of the noun.

Čim means as soon as.

It introduces something that happened immediately after something else:

• čim smo došli kući = as soon as we got home

It is stronger and more immediate than kad:

• kad = when
• čim = as soon as

So čim emphasizes that the cat removed the necklace right after they arrived home.

Because Croatian often leaves subject pronouns out when they are already clear from the verb.

• smo tells you it is we
• so mi does not have to be said

That is very normal in Croatian.

Also, notice the participle:

• došli = masculine plural
• došle = feminine plural

So smo došli usually means:

• a mixed group of people, or
• a group including at least one male, or
• sometimes just a neutral default plural

If the speakers were all female, it would normally be smo došle.

Because kući is a very common idiomatic form meaning home after verbs of motion.

So Croatian says:

• ići kući = go home
• doći kući = come home
• vratiti se kući = return home

This is different from being at home:

• kod kuće = at home

So:

• došli kući = came home
• bili kod kuće = were at home

That contrast is very useful to remember.