Ich mache Zusatzkosten geltend, denn ich habe die Belege.

Because geltend machen is a fixed verbal expression, and in a main clause the conjugated verb machen goes to the second position, while geltend stays at the end.

So:

• Ich mache ... geltend.
• not Ich geltend mache ...

This works a lot like a separable verb, even though geltend machen is usually treated as a verb + adjective expression.

Compare:

• Ich mache die Kosten geltend.
• Er macht seine Ansprüche geltend.

In a subordinate clause, the whole verbal part moves to the end:

• ..., weil ich Zusatzkosten geltend mache.

Geltend machen is a common formal expression meaning something like:

• to claim
• to assert
• to invoke
• to make a claim for

In this sentence, it means the speaker is officially claiming or submitting additional costs.

It is often used in formal, legal, insurance, or administrative contexts, for example with:

• Kosten geltend machen = to claim costs/expenses
• Ansprüche geltend machen = to assert claims/rights
• Schadensersatz geltend machen = to claim damages

So this is not casual everyday phrasing; it sounds fairly formal.

Zusatzkosten is a German compound noun:

• Zusatz = additional extra item, supplement
• Kosten = costs

German very often combines nouns into one word, so Zusatzkosten means additional costs or extra expenses.

This is very normal in German. English often uses separate words where German prefers compounds.

Other examples:

• Fahrtkosten = travel costs
• Nebenkosten = additional charges/utilities/service costs
• Anwaltskosten = lawyer’s fees/legal costs

German often omits the article when talking about something in a general or somewhat indefinite way, especially in formal styles.

So:

• Ich mache Zusatzkosten geltend.
  = I am claiming additional costs.
  This sounds general/formal.

But you could also say:

• Ich mache die Zusatzkosten geltend.
  = I am claiming the additional costs.
  This sounds more specific, referring to particular additional costs already known in the context.

So both are possible, but the version without the article is very natural in formal or bureaucratic language.

It is in the accusative case, because it is the direct object of machen in the expression etwas geltend machen.

You can think of the structure as:

• ich mache = I make / I claim
• was? = what?
• Zusatzkosten = additional costs

Since Zusatzkosten is plural, the form is the same in nominative and accusative:

• nominative plural: die Zusatzkosten
• accusative plural: die Zusatzkosten

So the case is accusative, even though the form itself does not change here.

Both denn and weil can mean because, but they behave differently.

1. With denn, normal word order stays the same:

• ..., denn ich habe die Belege.

Here, ich habe follows normal main-clause word order.

2. With weil, the verb goes to the end:

• ..., weil ich die Belege habe.

So the sentence could also be:

• Ich mache Zusatzkosten geltend, weil ich die Belege habe.

The difference is mostly about style and structure:

• denn is a coordinating conjunction
• weil is a subordinating conjunction

In many contexts, weil is more common in everyday speech, while denn can sound a bit more written or formal. In this sentence, denn fits the formal tone well.

Here, Belege means supporting documents, receipts, or proof showing that the extra costs are real.

In financial, legal, or administrative German, a Beleg is often:

• a receipt
• an invoice
• a voucher
• some document proving a payment or expense

So ich habe die Belege means the speaker has the documents needed to support the claim.

Die Belege refers to specific documents that are understood from the situation.

So:

• Ich habe die Belege.
  = I have the receipts/supporting documents.

This suggests the relevant documents are known or expected.

If you said:

• Ich habe Belege.

that would mean more generally:

• I have receipts/documents.

That is possible, but die Belege sounds more natural when talking about the exact documents needed to justify the claim.

Because after denn, German uses normal main-clause word order.

So the pattern is:

• denn + subject + verb
• denn ich habe die Belege

You would get habe ich only in a yes/no question or in certain inverted structures:

• Habe ich die Belege? = Do I have the receipts?

But after denn, there is no inversion.

It sounds fairly formal and somewhat bureaucratic/administrative.

The main reason is the expression Zusatzkosten geltend machen, which is typical of:

• insurance claims
• reimbursement requests
• legal writing
• formal complaints
• official correspondence

A more everyday way to say something similar might use a simpler verb such as beantragen or fordern, depending on context, but geltend machen is especially common when someone is asserting a right or submitting a formal claim.

Yes. It is a very common formal expression with several kinds of nouns, especially in legal or official language.

Examples:

• Ansprüche geltend machen = to assert claims/rights
• Rechte geltend machen = to assert rights
• Forderungen geltend machen = to make demands/claims
• Schadensersatz geltend machen = to claim damages
• Kosten geltend machen = to claim expenses

So once you learn geltend machen, it becomes useful in many formal contexts.