Amazed that windbag has to shut up (6)

I believe the answer is:

aghast

I'm a little stuck... Click here to teach me more about this clue!

'amazed' is the definition.
(I've seen this in another clue)

'windbag has to shut up' is the wordplay.
'AGHAST' is hidden in amongst the letters.
But, I am uncertain how this is indicated.

'that' is the link.

Can you help me to learn more?

(Other definitions for aghast that I've seen before include "Horrified, appalled" , "Deeply shocked" , "Filled with fear or horror" , "Horror-stricken, filled with consternation" , "Filled with horror or shock" .)

I've seen this clue in the Sydney Morning Herald.

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