To be clearer, if you replace extremely with "so", it would be 为什么你看起来那么生气? But both are alright for speaking.

I'm bilingual Chinese/English from my background, with no formal academic training in Chinese so take this with a grain of salt: as far as I can tell, it kind of forms a complex with 看 - strung together, 看起来 has the effect of making 看 reflexive, referring to the appearance of the person, instead of referring to the action of the person.

When using 看 and 起来 together, it expresses the meaning of inference about a state of someone or something, for instance, 这杯水看起来很烫= this glass of water seems very hot.

The literal meaning of 起来 is stand up(after someone falls), in colloquialism it can also mean get up.

看起来=seems to be 看=look/see

Something else I wanna share just came up . '很生气' '极生气' '非常(的)生气' '出离(的)愤怒' all mean 'extremely angry'

There are some subtle differences among these expressions. You can't add 'de' to these phrases: '很生气' '极生气', e.g. '很的生气' is grammarly incorrect.

But the expressions such as 非常的生气 are fine, and you could omit 'de' in these phrases.

It seems that the pattern is if the adverbial consists of two words(e.g. 非常or出离), then you could omit 'de' in the phrase, but if the adverbial consists of only one word(e.g. 很or极), then you can't add 'de' to the phrase.

Tbh, as a native chinese speaker, I'm not 100% confident this rule applies to every situation of the grammar pattern of 'adverbial+adjective' precisely.