Why 看 is used in 我看得目瞪口呆?
The English translation is not literal. The idiom 目瞪口呆 means something like 'eyes wide-staring, mouth stiff'. Of course that is a physical description of someone who is dumbfounded.
To be grammatical, you first need the verb 看 'to see; to look' in your sentence, followed by 得 which suffixes an adverb or an adverbial phrase (目瞪口呆) to describe the extent of shock resulting from your looking at something dumbfounding.
Other preceding verbs are also possible, such as 被嚇 'to be scared', as in 被嚇得目瞪口呆 (lit. to be scared, so much so one's eyes become wide-staring and mouth stiff). In this case, the speaker is not looking at anything.