To be surprised or astonished.
The British comedian Frankie Howerd used to say in mock astonishment: “I’m flabbergasted—never has my flabber been so gasted!”. That’s about as good an explanation for the origin of this word as you’re likely to get. It turns up first in print in 1772, in an article on new words in the Annual Register. The writer couples two fashionable terms: “Now we are flabbergasted and bored from morning to night”. (Bored—being wearied by something tedious—had appeared only a few years earlier.) Presumably some unsung genius had put together flabber and aghast to make one word.
The source of the first part is obscure. It might be linked to flabby, suggesting that somebody is so astonished that they shake like a jelly. It can’t be connected with flapper, in the sense of a person who fusses or panics, as some have suggested, as that sense only emerged at the end of the nineteenth century. But flabbergasted could have been an existing dialect word, as one early nineteenth-century writer claimed to have found it in Suffolk dialect and another—in the form flabrigast— in Perthshire. Further than this, nobody can go with any certainty.