Jeg skriver til kundeservice, fordi hjemmesiden viser, at lampen er udsolgt.

Because Danish usually writes compound nouns as a single word.

• kundeservice = kunde
  • service
• hjemmeside = a fixed compound meaning website

This is very common in Danish. Where English might sometimes use two words, Danish often joins them:

• computer screen → computerskærm
• customer service → kundeservice

So hjemmesiden is just hjemmeside with the definite ending added.

Because when you mean write to someone, Danish normally uses skrive til.

• Jeg skriver til kundeservice = I am writing to customer service
• Jeg skriver en e-mail = I am writing an email

So the person or department you are contacting is introduced with til.

You can also have both:

• Jeg skriver en e-mail til kundeservice = I am writing an email to customer service

Here, kundeservice is being used like a general department name, similar to English customer service.

So:

• Jeg skriver til kundeservice sounds natural = I’m writing to customer service

You could sometimes say kundeservicen if you mean a very specific, already identified customer-service department, but in this kind of sentence the bare form kundeservice is the most natural.

That -en is the Danish definite article attached to the noun. Instead of a separate word like English the, Danish often puts the definiteness at the end.

• en hjemmeside = a website

• hjemmesiden = the website

• en lampe = a lamp
• lampen = the lamp

Both hjemmeside and lampe are common gender nouns, so they take -en in the definite singular.

Yes. Danish often uses the simple present where English would use either the simple present or the present progressive.

So jeg skriver can mean:

• I write
• I am writing

The context tells you which meaning fits. In your sentence, I am writing is the natural English translation.

The same applies to the rest:

• hjemmesiden viser = the website shows / is showing
• lampen er udsolgt = the lamp is sold out

Fordi means because.

It introduces a subordinate clause:

• Jeg skriver til kundeservice, fordi ...
• I am writing to customer service, because ...

In Danish, subordinate clauses often have different word-order behavior from main clauses, especially when adverbs like ikke are involved.

For example:

• fordi hjemmesiden ikke viser det
• because the website does not show it

Notice that ikke comes before the verb viser in the subordinate clause.

In your sentence there is no adverb like ikke, so the word order looks fairly similar to English:

• fordi hjemmesiden viser, at ...

Here at means that and introduces the content of what the website shows.

• hjemmesiden viser, at lampen er udsolgt
• the website shows that the lamp is sold out

In English, that is often omitted:

• The website shows the lamp is sold out

In Danish, keeping at is very normal and often the safest choice, especially in writing.

Yes. Viser literally means shows, but in this context it can also be understood as indicates or even says.

So:

• hjemmesiden viser, at ... = the website shows / indicates that ...

This is a very natural way in Danish to talk about what a website, app, or screen displays.

Udsolgt means sold out, and it is used with er to describe the item’s current status.

• Lampen er udsolgt = The lamp is sold out

So this is not really an action here; it is a condition or state.

You will see the same pattern with many things:

• Koncerten er udsolgt = The concert is sold out
• Produktet er udsolgt = The product is sold out

This is the standard and natural way to say it.

Because the sentence refers to a specific lamp, not just any lamp.

Usually that means the speaker and listener already know which lamp it is, or it is clear from the context, for example from a product page.

Compare:

• en lampe = a lamp
• lampen = the lamp

If you were talking about an unspecified lamp in general, you would use en lampe instead.

Those commas reflect a common Danish punctuation style that marks the start of subordinate clauses.

So in your sentence:

• comma before fordi
• comma before at

This is very normal to see in Danish.

You may also come across Danish written without those commas if the writer is using the no-start-comma system. So punctuation can vary a bit, but the version you have is completely standard and learner-friendly because it clearly marks the clause structure.