De verjaardagstaart is voor de jarige.

Because the main noun in the compound is taart, and taart is a de-word in Dutch.

In Dutch compound nouns, the last part determines the grammatical gender and article. So:

• taart = cake, pie → de taart
• verjaardagstaart = birthday cake → de verjaardagstaart

So even though verjaardag is a separate noun on its own, the whole compound takes its article from taart.

It is a compound noun made from:

• verjaardag = birthday
• taart = cake

So:

• verjaardag + taart = verjaardagstaart

Dutch very often joins nouns together into one word where English would use two separate words. That is why birthday cake becomes verjaardagstaart, not two separate words.

Here, voor means for in the sense of intended for or meant for.

So:

• De verjaardagstaart is voor de jarige. = The birthday cake is for the birthday person.

Be aware that voor can have other meanings too, depending on context, such as:

• before
• in front of
• for

But in this sentence, the meaning is clearly for.

De jarige means the person whose birthday it is.

Dutch often turns an adjective into a noun when talking about a person. Here:

• jarig = having a birthday / celebrating one’s birthday
• de jarige = the birthday person

A very natural English translation is:

• the birthday boy/girl
• the birthday person

depending on context.

Because jarige here is an adjective being used like a noun, and in that position Dutch normally uses the -e form.

Compare:

• Hij is jarig. = He is having his birthday.
  Here jarig is just an adjective after is.

• De jarige krijgt cadeaus. = The birthday person gets presents.
  Here jarige is being used as a noun-like word, so the -e form appears.

This is very common in Dutch:

• de zieke = the sick person
• de bekende = the famous person / celebrity
• de jarige = the birthday person

Yes. This is a completely normal main clause:

• De verjaardagstaart = subject
• is = verb
• voor de jarige = prepositional phrase

So the structure is basically:

• Subject + verb + complement

That is standard Dutch main-clause word order.

Because de jarige functions as a noun phrase meaning the birthday person.

When Dutch uses an adjective as a noun for a person, it often takes de:

• de oude = the old one / old person
• de rijke = the rich one / rich person
• de jarige = the birthday person

So de is not random here; it is part of making jarige work as a noun-like expression.

Yes. De jarige is gender-neutral in meaning.

It can refer to:

• a boy
• a girl
• a man
• a woman

The exact person is understood from context. If you want to be more specific, Dutch can also say things like:

• de jarige jongen = the birthday boy
• de jarige meid = the birthday girl

But de jarige by itself is very common and natural.

Yes. Both nouns are definite here:

• de verjaardagstaart = the birthday cake
• de jarige = the birthday person

So the speaker is talking about a specific cake and a specific person, not just any birthday cake or any birthday person.

If you wanted an indefinite version, it would look different, for example:

• Een verjaardagstaart is voor de jarige. = A birthday cake is for the birthday person.

But in real life, the definite version sounds very natural when everyone knows which cake and which person are being talked about.

You could use that as a learning aid, but it is not natural English.

A more natural translation is:

• the birthday person
• the person whose birthday it is

So literally, Dutch is doing something a bit like the birthday one, but idiomatic English usually avoids that wording.

A rough English-friendly pronunciation is:

• duh vər-YAAR-dakh-staart is voor duh YAA-ri-guh

A few helpful notes:

• verjaardag has the strong stress on jaar
• g in standard Dutch is a rough throat sound, especially in verjaardag
• jarige ends with a soft -uh sound in the final -e

The exact pronunciation depends somewhat on region, but that approximation is good enough to get started.