Den røde kjole passer til det sorte bælte.

Because Danish nouns have grammatical gender.

• kjole is common gender, so it takes den in the definite phrase
• bælte is neuter, so it takes det

You can see the same difference in the indefinite forms:

• en kjole = a dress
• et bælte = a belt

So:

• den røde kjole = the red dress
• det sorte bælte = the black belt

Because in Danish, when an adjective comes before a definite noun, the adjective usually takes the -e form.

So you get:

• en rød kjole = a red dress
• den røde kjole = the red dress

and

• et sort bælte = a black belt
• det sorte bælte = the black belt

This -e form is very common in Danish. It is used with definite nouns and also in many plural phrases.

Because Danish usually marks definiteness differently when there is an adjective before the noun.

Compare:

• kjolen = the dress
• bæltet = the belt

But with an adjective:

• den røde kjole = the red dress
• det sorte bælte = the black belt

So with a preceding adjective, Danish normally uses:

den/det/de + adjective + noun

and the noun stays in its basic form.

Here, passer til means matches, goes with, or suits.

So the sentence means that the red dress and the black belt look good together.

That is important because passer can mean different things in different contexts. For example:

• Kjolen passer. = The dress fits.
• Kjolen passer til bæltet. = The dress matches the belt.

So in this sentence, it is about matching, not physical size.

Because passe til is the normal Danish expression for saying that one thing suits or matches another.

So:

• A passer til B = A goes with B / A matches B

This is especially common with:

• clothes
• colors
• furniture
• decorations

For this sentence, til is not being used in a very literal English-style to sense. It is part of the natural Danish pattern.

The sentence has a very normal Danish main-clause order:

• Den røde kjole = subject
• passer = verb
• til det sorte bælte = prepositional phrase

So the structure is basically:

subject + verb + rest

That is similar to English here.

A useful extra note: Danish main clauses follow a verb-second pattern, so the finite verb normally comes early in the sentence.

It refers to specific items, because it uses the definite forms:

• den røde kjole = the red dress
• det sorte bælte = the black belt

If you wanted to talk more generally, you would normally use the indefinite forms:

• En rød kjole passer til et sort bælte.
• A red dress goes with a black belt.

So the original sentence sounds like you already know which dress and which belt are being talked about.

It would be:

En rød kjole passer til et sort bælte.

That means:

A red dress matches a black belt.

This is a very useful comparison:

• en rød kjole -> den røde kjole
• et sort bælte -> det sorte bælte

So you can see both the article change and the adjective change.

Not in this sentence.

Here, den and det are acting as definite articles, so they mean the.

They can sometimes also be used in other ways in Danish, including as pronouns, but in:

• den røde kjole
• det sorte bælte

they simply mean the.

So a learner should understand them here as part of the pattern for definite noun phrases with adjectives.