Min søster finder et sort bælte, som passer til hendes kjole.

Because søster is a common-gender noun in Danish. Its basic form is en søster, so the correct possessive is min.

A quick pattern:

• min for common-gender singular nouns
• mit for neuter singular nouns
• mine for plural nouns

Examples:

• min søster
• mit hus
• mine søstre

Because bælte is a neuter noun, so it takes et.

In Danish, nouns have grammatical gender, and you usually have to learn that together with the noun:

• en søster
• et bælte
• en kjole

That is why the sentence has et sort bælte.

Danish adjectives agree with the noun they describe.

A useful basic pattern is:

• common-gender singular indefinite: often the basic form
• neuter singular indefinite: often -t
• plural or definite: usually -e

With sort, the form looks the same in both common-gender and neuter singular:

• en sort bil
• et sort bælte
• sorte biler

So in et sort bælte, sort is the correct form.

Because hendes already functions as the determiner.

In Danish, as in English, you normally do not use en/et together with a possessive:

• hendes kjole = her dress
• not hendes en kjole

So once you have hendes, no separate article is needed.

Som introduces a relative clause. It refers back to et sort bælte and adds more information about it.

So the structure is:

• et sort bælte
• som passer til hendes kjole

In English, som here often corresponds to that or which.

Because passe til is a very common Danish expression meaning to match, to go with, or to suit something.

So:

• passer til hendes kjole = matches her dress / goes with her dress

This is a good example of a verb plus preposition that should be learned as a unit:

• passe til noget

This is a very important Danish grammar point.

Sin/sit/sine refers back to the subject of the same clause.

In the relative clause:

• som passer til hendes kjole

the subject is som, which refers to bælte. So if you used sin, it would grammatically point back to the belt, not to the sister.

That is why hendes is correct here: it refers to the sister, who is not the subject of that clause.

A helpful contrast:

• Min søster finder et bælte til sin kjole
  Here sin can refer to min søster, because she is the subject of that clause.

• Min søster finder et bælte, som passer til hendes kjole
  Here the subject inside the relative clause is som = the belt, so hendes is needed.

Not all Danish possessives behave the same way.

Some possessives change form:

• min / mit / mine
• din / dit / dine
• sin / sit / sine

But others do not change:

• hans
• hendes
• deres
• vores
• jeres

So you get:

• hendes kjole
• hendes hus
• hendes sko

The form hendes stays the same no matter what noun follows.

The main clause has normal Danish subject–verb–object order:

• Min søster = subject
• finder = verb
• et sort bælte = object

Then comes the relative clause:

• som passer til hendes kjole

Inside that clause, som is the subject, followed by the verb passer.

So the overall structure is:

• main clause: Min søster finder et sort bælte
• relative clause: som passer til hendes kjole

Both are in the present tense.

In Danish, the present tense is often formed with -r:

• finder
• passer

Like English present tense, this can describe:

• something happening now
• a general fact
• a habitual action

Danish does not usually have a separate form exactly like English is finding or is matching, so context does a lot of the work.

It depends on the comma system being used.

In Danish, many writers use a comma before a subordinate or relative clause:

• Min søster finder et sort bælte, som passer til hendes kjole.

But in the new comma system, that comma can often be omitted:

• Min søster finder et sort bælte som passer til hendes kjole.

So the version with the comma is very common and fully correct, but you may also see the sentence written without it.