Moj brat se smijao kad je majmun uzeo bananu iz ruke jedne djevojčice.

Because the verb is smijati se, which means to laugh. In Croatian, se is part of this verb's normal form. It does not literally mean himself here.

So:

• smijati se = to laugh
• Moj brat se smijao = My brother was laughing / laughed

Many Croatian verbs are learned together with se, so it is best to memorize them that way.

This is a very common learner question. In standard Croatian, with se verbs in the 3rd person singular past, the auxiliary je is normally not used here.

So you say:

• On se smijao.
• Moj brat se smijao.

Not:

• On se je smijao.

Compare that with a non-se verb, where je does appear:

• Majmun je uzeo bananu.

So the contrast in your sentence is normal:

• brat se smijao
• majmun je uzeo

Because je is a clitic. Croatian clitics usually go in the second position of their clause.

In the clause:

• kad je majmun uzeo bananu

the word kad comes first, so the clitic je goes right after it.

This is why you get:

• kad je majmun uzeo...

rather than the more English-like order:

• kad majmun je uzeo... ← not the normal Croatian order

These are past participle forms used to make the Croatian past tense.

A very important point: in Croatian, these forms agree with the subject in gender and number.

Here both subjects are masculine singular:

• brat → smijao
• majmun → uzeo

If the subject were feminine, the form would change:

• Moja sestra se smijala.
• Majmunica je uzela bananu.

So English past tense does not change for gender, but Croatian past tense does.

This is about aspect, which is one of the biggest differences between English and Croatian.

• uzeti = perfective, a completed single action
• uzimati = imperfective, ongoing/repeated action

In this sentence, the monkey did one complete action: it took the banana. So uzeo is the natural choice.

At the same time:

• smijao se is imperfective and gives the background action
• uzeo is perfective and gives the specific event

So the sentence presents:

• the brother was laughing
• when the monkey took the banana

That imperfective + perfective combination is very common in Croatian storytelling.

Because bananu is the accusative singular, and here banana is the direct object of uzeo.

Dictionary form:

• banana = nominative singular

Direct object form:

• bananu = accusative singular

For many feminine nouns ending in -a, the accusative singular ends in -u:

• žena → ženu
• kuća → kuću
• banana → bananu

So uzeo bananu means took a/the banana.

Because the preposition iz takes the genitive case.

Dictionary form:

• ruka = hand

After iz:

• iz ruke = from/out of the hand

So:

• iz ruke = from the hand
• iz kuće = out of the house
• iz škole = from school

This is a fixed case pattern you simply have to learn with the preposition.

Because this whole phrase is affected by the noun ruke:

• iz ruke jedne djevojčice

Literally: from the hand of one/a certain little girl

Here djevojčice is in the genitive singular because it depends on ruke in a possessive relationship:

• ruka djevojčice = the hand of the girl

And jedne must match djevojčice in case, gender, and number.

So:

• jedna djevojčica = nominative
• jedne djevojčice = genitive

Croatian has no words exactly like English a and the. But jedan / jedna / jedno can sometimes do part of the job of a/an, especially when introducing someone or something not previously identified.

So:

• jedne djevojčice can mean a girl, one girl, or a certain girl, depending on context

In this sentence, it makes the girl sound indefinite or newly introduced.

Without jedne, the sentence would still be grammatical:

• iz ruke djevojčice

But jedne djevojčice sounds a bit more like English from the hand of a little girl.

Yes. Both mean when.

• kad = shorter, very common, natural in everyday language
• kada = longer, sometimes a bit more formal or emphatic

So in this sentence, kad is completely normal and natural.

You could also say:

• Moj brat se smijao kada je majmun uzeo bananu iz ruke jedne djevojčice.

The meaning is the same.

To some extent, yes. Croatian word order is more flexible than English word order. But the clitics still have placement rules, and some orders sound more natural than others.

The given sentence is neutral and natural:

• Moj brat se smijao kad je majmun uzeo bananu iz ruke jedne djevojčice.

You could move things for emphasis, for example:

• Kad je majmun uzeo bananu iz ruke jedne djevojčice, moj brat se smijao.

That shifts the focus a bit, but the basic meaning stays the same.

What you usually cannot do freely is place clitics like se and je anywhere you want. Their position is much more restricted than ordinary words.