geuphal ttae bisanggureul mos chajeulkka bwa yeonghwagwane deureogajamaja annaepaneul bwasseo.

It means for fear that, in case, or because I was worried that.

So 비상구를 못 찾을까 봐 means:

• because I was worried I might not be able to find the emergency exit
• in case I couldn't find the emergency exit

This pattern is very common when someone does something as a precaution.

A similar example:

• 비가 올까 봐 우산을 가져왔어.
  • I brought an umbrella because I was worried it might rain.

In your sentence, looking at the sign was the precaution.

Because the speaker is talking about a possible future problem from the point of view of that moment.

At the time they looked at the sign, the problem had not happened yet. They were thinking:

• What if I can't find the emergency exit later?

So Korean uses the future/possibility form:

• 찾을까 봐 = in case I might have to look / because I worry I might...
• 못 찾을까 봐 = because I worry I might not be able to find it

The main verb shows the actual tense of the whole sentence:

• 봤어 = looked / saw (past)

So the structure is basically:

• Worried about a possible later problem
  • did something in the past

못 찾다 means to be unable to find or to fail to find.

Here:

• 찾다 = to find
• 못 찾다 = to not be able to find

This 못 is the adverb meaning cannot / unable to.

So:

• 비상구를 찾다 = to find the emergency exit
• 비상구를 못 찾다 = to not be able to find the emergency exit

This is different from 안 찾다, which would mean not look for or not find because you didn't do it, depending on context.

Here it most naturally means when things become urgent, when I’m in a rush, or more specifically in an emergency situation.

Because the sentence talks about an emergency exit, 급할 때 is best understood as:

• when it becomes urgent
• in an emergency
• when I need it quickly

So it is not just ordinary being busy with work. The context pushes it toward a situation where I may suddenly need the exit.

-자마자 means as soon as.

So:

• 들어가다 = to enter
• 들어가자마자 = as soon as (someone) entered

In the sentence:

• 영화관에 들어가자마자 안내판을 봤어
• I looked at the sign as soon as I entered the movie theater

It emphasizes that there was almost no delay between entering and looking at the sign.

Both can connect two actions, but the nuance is different.

• -자마자 = immediately as soon as
• -고 나서 = after doing, with no strong sense of immediacy

Compare:

• 영화관에 들어가자마자 안내판을 봤어.

  • As soon as I entered the theater, I looked at the sign.
  • Strong feeling of right away

• 영화관에 들어가고 나서 안내판을 봤어.

  • After entering the theater, I looked at the sign.
  • More neutral; there could have been a short gap

So -자마자 fits well because the speaker wanted to check the sign immediately.

Because 들어가다 normally takes a destination marked by 에.

• 영화관에 들어가다 = to go into the movie theater

Here 에 marks the place entered.

So:

• 영화관에 들어가자마자 = as soon as I entered the movie theater

In standard Korean, 에 들어가다 is the normal pattern.

Because each one is the direct object of a different verb.

• 비상구를 못 찾다
  • 비상구를 is the thing that might not be found
• 안내판을 봤어
  • 안내판을 is the thing that was looked at

So the sentence contains two separate verb phrases:

1. 비상구를 못 찾을까 봐

   • because I was worried I might not find the emergency exit

2. 안내판을 봤어

   • I looked at the sign

Each verb has its own object, so both can take 를/을.

안내판 means something like:

• information sign
• guide sign
• directional sign
• signboard

In this sentence, it probably means a sign that helps people locate things in the theater, including the emergency exit.

So it does not necessarily mean the actual illuminated EXIT sign above a door. It could be a larger guide map or directional board near the entrance.

봤어 is the casual informal past form of 보다 (to see / to look at).

• Dictionary form: 보다
• Past casual: 봤어
• Past polite: 봤어요

So:

• 안내판을 봤어 = casual
• 안내판을 봤어요 = polite

Also, 봤어 comes from a contraction of 보았어.

Yes, the subject is omitted, which is very normal in Korean.

The understood subject is most naturally I:

• (나는) 급할 때 비상구를 못 찾을까 봐 영화관에 들어가자마자 안내판을 봤어.

English usually wants the subject stated, but Korean often leaves it out when it is obvious from context.

So the same person:

• might not find the emergency exit later,
• enters the theater,
• and looks at the sign.

Here it means look at, not just see accidentally.

Because the speaker checked the sign on purpose, 안내판을 봤어 is best understood as:

• I looked at the sign
• I checked the sign

So even though 보다 can mean both see and look at, the context here clearly suggests a deliberate action.

Yes. That is one of the most important things this sentence expresses.

The flow is:

1. There might be a problem later

   • I might not be able to find the emergency exit when it becomes urgent

2. So I take preventive action now

   • I looked at the sign as soon as I entered

That is exactly the kind of meaning -을까 봐 / -ㄹ까 봐 often gives:
worry about a possible future situation leading to a precaution now.