Страховое возмещение выплатят при условии, что заявление о страховой выплате проверено.

This is a very common Russian pattern. Выплатят literally means they will pay out, but Russian often uses 3rd person plural with no subject when the doer is unknown, unimportant, or obvious from context.

So here:

• Страховое возмещение выплатят
  = The insurance compensation will be paid out = literally, They will pay out the insurance compensation

In English, this is often translated with a passive: will be paid.

This structure is especially common in official, bureaucratic, and impersonal language.

Both are insurance-related terms, and in many contexts they are very close in meaning.

• страховое возмещение = insurance compensation / indemnity
• страховая выплата = insurance payment / payout

In this sentence, they appear together, which can feel confusing:

• Страховое возмещение выплатят...
• ...заявление о страховой выплате проверено

A rough distinction is:

• страховое возмещение focuses on the compensation itself
• страховая выплата focuses on the act or result of paying under the insurance policy

In official language, Russian often uses near-synonymous legal terms like this. A learner does not need to assume there is always a big practical difference in every sentence.

This is a fixed expression:

• при условии = under the condition
• что = that

Together, при условии, что means:

• provided that
• on condition that
• if, but only if

It is more formal than plain если (if).

Compare:

• если заявление проверено = if the application has been checked
• при условии, что заявление проверено = provided that the application has been checked

So this phrase is especially typical in contracts, legal texts, instructions, and formal notices.

Because что introduces a subordinate clause.

The sentence has two parts:

1. Страховое возмещение выплатят
2. заявление о страховой выплате проверено

These are joined by при условии, что, and Russian punctuation normally separates the main clause from the subordinate clause with a comma:

• ..., что ...

So the comma is required here.

Проверено is a short-form passive participle, and it agrees with заявление.

• заявление = application, statement
• It is neuter singular

So the participle must also be neuter singular:

• заявление проверено = the application has been checked / verified

Compare:

• документ проверен — masculine singular
• заявление проверено — neuter singular
• заявка проверена — feminine singular
• документы проверены — plural

This short form is very common in formal written Russian for states or results:

• подписано
• одобрено
• проверено
• отправлено

Both о and на can appear with заявление, but they do not always mean exactly the same thing.

Here:

• заявление о страховой выплате = an application / statement concerning an insurance payment

This is a formal, bureaucratic way to name the document.

You may also hear:

• заявление на выплату
• заявление на страховую выплату

That often means an application for payment in a more practical sense.

Very roughly:

• о + prepositional often sounds like regarding / concerning
• на + accusative often sounds like for / requesting

In official Russian, заявление о страховой выплате is a natural document-style phrase.

Because the noun выплата comes after the preposition о, and о usually takes the prepositional case.

Base form:

• страховая выплата

After о:

• о страховой выплате

The adjective and noun both change:

• страховая → страховой
• выплата → выплате

So:

• заявление о страховой выплате = application concerning an insurance payment

It is in the accusative case, because it is the direct object of выплатят.

• выплатят что? — страховое возмещение

However, страховое возмещение is neuter and inanimate, so its accusative form looks exactly like its nominative form. That is why it may not look as if anything changed.

So:

• nominative: страховое возмещение
• accusative: страховое возмещение

Same form, different function.

Sometimes yes, depending on style.

Literally, the Russian says:

• They will pay out the insurance compensation

Natural English often prefers:

• The insurance compensation will be paid
• You will receive the insurance compensation

The Russian sentence itself does not explicitly say you. It keeps the wording impersonal and official. That is very typical of notices, policies, and legal language.

So you will receive may be a natural translation in context, but it is not the literal grammatical structure of the Russian.

Russian often uses a short-form passive participle by itself to describe a completed state:

• заявление проверено = the application has been checked / is checked

You do not need быть in the present tense. In modern Russian, present-tense быть is usually omitted.

Compare:

• заявление проверено = the application is/has been checked
• заявление было проверено = the application was checked / had been checked
• заявление будет проверено = the application will be checked

So проверено alone is the normal present-result form here.

It is definitely formal and administrative.

Clues include:

• страховое возмещение
• при условии, что
• заявление о страховой выплате
• the impersonal выплатят
• the short-form passive проверено

In everyday speech, a person might say something simpler, for example:

• Вам выплатят деньги, если заявление проверили.
• Выплату сделают после проверки заявления.

But in official texts, the original sentence sounds normal and natural.