Could someone tell me why is よう being used in this sentence?

Question

From the context I think he meant: Why was that person so scared?

But what does this よう mean? Couldn't it be just 怯え?

あの怯えようは何だ？

Answer 1

This kind of よう after masu-stem forms a noun phrase meaning "the way of ～", "how one ～", "one's manner of ～", etc. I think it's a relatively uncommon, literary usage.

あの怯えようは何だ？
(lit.) What's his way of being frightened? / Hey look how he's frightened!

よう may appear in the following forms:

• dictionary-form + 様【よう】: so that; in order to (泳ぐよう, 見るよう, 食べるよう) Usu. in kana
• masu-stem + 様【よう】: way of ～ (泳ぎよう, 見よう, 食べよう) Usu. in kana
• As part of the volitional-form of vowel-stem verbs: Let's ～ ; I think I will ～ (見よう, 食べよう)
• dictionary-form + 用【よう】: for the purpose of ～ (泳ぐ用, 見る用, 食べる用)

Examples:

• 見ようによっては、正しいと言える。
  Depending on how you see it, you can say it's correct.
• 馬鹿とはさみは使いよう (Japanese proverb)
  (lit.) As for idiots and scissors, it's all about how you use them.
  Everything comes in handy when used right.

masu-stem + っぷり is a similar pattern which emphasizes "how {extraordinary/intensively/etc} one does something". (See this)

• あの怯えっぷりは何だ？　≒　あの怯えようは何だ？
• 彼の飲みっぷりは素晴らしい。 I'm impressed that he's a good drinker.

怯え in its own right means "fear" (noun), which is not something one can directly see. So you need よう/っぷり/かた after 怯え.

Answer 2

I'm not sure where you read that and in what context it is used but after a few searches on the internet, I can say that the usage of 怯えよう is "nearly" nonexistent.

よう could be 様 or 用.
If it is 用, the sentence could mean something like "whats the point of getting scared?" or "what's the reason you are getting scared for?"

If it is 様, it could mean "what's up with that scared look on your face?" etc.