baeteoriga eobseoseo hyudaeponi kkeojyeosseoyo.

없어서 comes from 없다 + -아서/어서.

• 없다 = to not exist, to not be there, or to not have
• -아서/어서 = a connector meaning because, so, or and therefore

So 배터리가 없어서 means because there was no battery / because the battery was gone / because there was no battery power.

In this sentence, it gives the reason for what happened next.

Because the sentence has two linked clauses, and each clause has its own subject:

• 배터리가 없어서 = the battery was not there / had run out
• 휴대폰이 꺼졌어요 = the cell phone turned off

So Korean is not using two subjects for one verb. It is using one subject in the reason clause and another subject in the main clause.

This is very normal in Korean.

Literally, yes, it can mean there was no battery.

But in everyday Korean, when talking about phones or devices, 배터리가 없다 is often understood more loosely as:

• the battery is dead
• there’s no battery left
• the charge ran out

So in this sentence, most people would naturally understand it as the phone turned off because the battery ran out, not necessarily that someone removed the battery.

This is an important difference:

• 꺼지다 = to go out, to turn off, to become off
  
  • 꺼졌어요 = it turned off
• 끄다 = to turn something off
  
  • 껐어요 = someone turned it off

So:

• 휴대폰이 꺼졌어요 = the phone turned off
• 휴대폰을 껐어요 = someone turned off the phone

Here, the phone shut off because of the battery, so 꺼졌어요 is the natural choice.

The dictionary form is 꺼지다.

It is built like this:

• 꺼지다 = to turn off / go off
• 꺼졌- = past tense form
• -어요 = polite ending

So 꺼졌어요 means turned off in polite past tense.

Because Korean often does not mark the reason clause separately for past tense when the main clause already makes the whole situation clearly past.

So:

• 배터리가 없어서 휴대폰이 꺼졌어요 = natural
• 배터리가 없었어서 휴대폰이 꺼졌어요 = awkward and unusual

In other words, Korean often lets the final verb carry the tense for the whole sentence.

이/가 is the neutral subject marker, so it fits well when you are simply stating what happened.

• 배터리가 없어서 휴대폰이 꺼졌어요 sounds like a straightforward explanation.

If you used 은/는, it would add a topic or contrast feeling:

• 배터리는 없어서 휴대폰이 꺼졌어요
• 휴대폰은 꺼졌어요

Those are possible in some contexts, but they sound more marked, as if you are contrasting the battery or the phone with something else.

For a simple factual sentence, 이/가 is the most natural choice.

Yes, very often.

If the context already makes it obvious what turned off, Korean can drop it:

• 배터리가 없어서 꺼졌어요.

This would still naturally mean It turned off because the battery ran out.

Korean frequently omits subjects and objects when they are understood from context. Including 휴대폰이 just makes the sentence more explicit.

They are very close in meaning.

• 휴대폰 = mobile phone, cell phone
• 핸드폰 = a very common everyday word for cell phone
• 스마트폰 = smartphone

In many situations, 휴대폰 and 핸드폰 can both be used.
휴대폰 feels a bit more standard or neutral, while 핸드폰 is very common in casual speech.

So this sentence could also be said as:

• 배터리가 없어서 핸드폰이 꺼졌어요.

꺼졌어요 is in the standard polite style.

That means it is appropriate for:

• everyday conversation
• speaking politely to someone you do not know well
• many normal social situations

It is less formal than 꺼졌습니다, but more polite than casual forms like 꺼졌어.

So the sentence sounds natural, polite, and conversational.