In a manner that shows amazement, surprise, or astonishment; in an amazed way.
From amazed (the adjective form of amaze, from Old English 'amasian' meaning to confuse) plus '-ly' to create an adverb. The '-ed' participle plus '-ly' is a standard English adverbial pattern.
Adverbs like this show English's flexibility—we can take almost any feeling and instantly convert it into a description of how someone does something, which is why English speakers can express shades of emotion other languages might need entire phrases to convey.
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