L’assuré appelle l’assureur au cas où le dommage augmenterait.

They are two different people or parties in the insurance relationship:

• l’assuré = the insured person, the person who is covered
• l’assureur = the insurer, the company or entity providing the insurance

They look very similar, so they are easy to confuse. A useful memory trick is:

• -é → the person insured
• -eur → the one who does the insuring

In legal or insurance French, these terms are often used as role labels, which is why you see the insured and the insurer rather than names.

Because French uses elision before a vowel sound.

The definite articles le and la become l’ before a noun beginning with a vowel or a mute h:

• le assuré → l’assuré
• le assureur → l’assureur

So l’ here just means the.

Because appelle is the conjugated verb, while appeler is the infinitive.

The infinitive is appeler = to call.
In this sentence, the subject is l’assuré, so the verb must be conjugated in the present tense:

• j’appelle
• tu appelles
• il / elle appelle

So l’assuré appelle means the insured calls.

A useful spelling point: appeler is one of those verbs that doubles the l in some present-tense forms, which is why you get appelle.

Because appeler takes a direct object in French.

You say:

• appeler quelqu’un = to call someone

So:

• L’assuré appelle l’assureur
  = The insured calls the insurer

You do not put à before l’assureur here.

Au cas où is a fixed expression meaning in case.

It introduces a possible situation that may happen. In this sentence, it means something like:

• in case the damage increases
• in case the damage were to increase

It is best learned as a single chunk: au cas où.

Because au is part of the fixed expression.

Grammatically, au is the contraction of à + le, but in practice learners should treat au cas où as a set phrase meaning in case.

So even though you can analyze it, the most useful thing is simply to memorize:

• au cas où + clause

Example:

• Au cas où il pleuvrait... = In case it rains / were to rain...

Because after au cas où, standard French normally uses the conditional, not the present indicative.

So:

• au cas où le dommage augmenterait

means that the increase in damage is only a possibility, not a fact.

This is one of the most important grammar points in the sentence. In English, we usually just say in case the damage increases, but French often uses the conditional here.

Because au cas où is one of those expressions that normally takes the conditional in standard French.

Many English speakers expect the subjunctive after expressions involving uncertainty, but this is not one of those cases. So standard French prefers:

• au cas où il viendrait
• au cas où le dommage augmenterait

not the subjunctive.

So this is a pattern worth memorizing:

• au cas où + conditional

Here, dommage means damage, harm, or loss, especially in a legal or insurance context.

That is different from the everyday exclamation:

• Dommage ! = Too bad!

In this sentence, it is a technical or formal noun. It refers to the damage that has occurred and might get worse.

This is a very common question.

• au cas où is followed by a clause with a verb
• en cas de is followed by a noun

So:

• au cas où le dommage augmenterait
  = in case the damage were to increase

But:

• en cas de dommage
  = in case of damage

A simple rule:

• en cas de + noun
• au cas où + verb

Because the main clause and the subordinate clause are doing different jobs.

• L’assuré appelle l’assureur describes the action or procedure
• au cas où le dommage augmenterait gives the hypothetical situation that might trigger that action

The present tense in the main clause can sound like a general rule, instruction, or standard procedure:

• The insured calls the insurer if/in case...

So the sentence is not necessarily talking about one exact moment right now. It can describe what one does in that kind of situation.

Because où and ou are different words in French:

• où = where
• ou = or

In au cas où, the correct spelling is où with a grave accent.

The accent helps distinguish the two words in writing, even though they are usually pronounced the same.

Yes, s’aggraverait is also very natural, and in many contexts it may sound more idiomatic.

• le dommage augmenterait = the damage would increase
• le dommage s’aggraverait = the damage would worsen / get more serious

The sentence you were given is grammatical and understandable. In insurance or legal wording, augmenter can make sense if the amount or extent of the damage becomes greater. But in everyday French, many speakers might also choose s’aggraver for the idea of damage getting worse.