sensei ha zisin no ato sugu ni hizyouguti kara hinansuru you ni iimasita.

Because あと is a noun meaning after or the time after. In Japanese, when one noun modifies another noun, の is often used.

So:

• 地震 = earthquake
• あと = after / the time after
• 地震のあと = after the earthquake

It is literally something like the after of the earthquake.

は marks the topic: as for the teacher, ...

So 先生は sets up the sentence as being about the teacher and then tells you what the teacher did.

If you used 先生が, it would sound more like you are identifying the teacher as the one who said it, often with a bit more emphasis or contrast, such as:

• 先生が言いました = The teacher said it

In this sentence, は is the most natural neutral choice.

They both mean immediately or right away, and in many cases they are interchangeable.

• すぐ = immediately
• すぐに = immediately

The に often makes it sound a little more adverb-like or slightly smoother in formal written speech, but the meaning here is basically the same.

So both of these are natural:

• 地震のあとすぐ避難する
• 地震のあとすぐに避難する

から marks a starting point, source, or route.

Here, 非常口から避難する means evacuate through/from the emergency exit.

So:

• 非常口 = emergency exit
• から = from / through
• 非常口から = through the emergency exit

A learner may expect に, but に usually marks a destination, and that does not fit here.
非常口に避難する would sound like evacuate to the emergency exit, which is not the intended meaning.

A natural alternative would be:

• 非常口を使って避難する = evacuate using the emergency exit

This is the pattern:

• dictionary form + ように言う

It means:

• to tell someone to do something
• to instruct someone to do something

So:

• 避難するように言いました = said/told [someone] to evacuate

The ように here is part of a set expression with 言う. It turns the action into the content of the instruction.

Other examples:

• 早く行くように言いました = I told him to go early.
• 忘れないように言いました = I told them not to forget.

Because that is the normal form used in this pattern.

With ように言う, Japanese usually uses:

• verb dictionary form + ように言う
• verb negative form + ように言う

Examples:

• 行くように言う = tell someone to go
• 食べるように言う = tell someone to eat
• 遅れないように言う = tell someone not to be late

So 避難するように言いました is exactly the expected form.

Because Japanese often leaves out information that is obvious from context.

In English, we often want an object like the teacher told the students. In Japanese, that person can be omitted if it is already understood.

So this sentence does not explicitly say who was told, but it could be understood as students, people nearby, or the listeners in general.

If you want to say it clearly, you can add it:

• 先生は生徒に、地震のあとすぐに非常口から避難するように言いました。
  The teacher told the students to evacuate through the emergency exit immediately after the earthquake.

Because this sentence is not using a direct quotation.

Japanese often uses と言う when reporting exact words:

• 非常口から避難してください と言いました
  = [Someone] said, Please evacuate through the emergency exit

But here the sentence is summarizing the instruction, not quoting the exact words. That is why it uses:

• 避難するように言いました

So:

• と言いました = said these words
• ように言いました = told/instructed [someone] to do something

It is the same form, but the meaning depends on the pattern.

Here:

• 避難するように言いました

means:

• told [someone] to evacuate

But in a sentence like:

• 忘れないようにメモしました

ように means:

• so that I won’t forget

So you should read ように together with the verb that follows it. In this sentence, the key pattern is ように言う.

Normally, it is understood as describing 避難する.

So the meaning is:

• after the earthquake, evacuate immediately through the emergency exit

That is the most natural reading because 地震のあとすぐに forms a strong time expression.

If you wanted to make it clearer that the teacher spoke immediately, you would probably rewrite the sentence in a different way.

Yes, but it would become a different structure.

避難してください is the actual request or instruction itself:

• 非常口から避難してください = Please evacuate through the emergency exit.

If you want to report that exact wording, you would say:

• 先生は非常口から避難してくださいと言いました。

But 避難するように言いました does not quote the exact words. It gives an indirect report: the teacher told people to evacuate.

So the difference is roughly:

• 避難してくださいと言いました = quoted command/request
• 避難するように言いました = reported instruction

A helpful breakdown is:

• 先生は — topic, as for the teacher
• 地震のあと — time phrase, after the earthquake
• すぐに — adverb, immediately
• 非常口から — route/source, through the emergency exit
• 避難するように — content of the instruction, to evacuate
• 言いました — main verb, said / told

Japanese usually puts the main verb at the end, so the sentence builds up all the details first and finishes with 言いました.