From the Merriam-Webster dictionary
quote as saying : to report or say (the exact words) of (someone) —usually used in the form be quoted as
He was quoted as saying there would be further delays.
"If they're flexible, we're flexible," the official was quoted as saying.
I guess there are three situations where the idiom would be used.
- the speaker that is quoted is famous
- the speaker that is quoted is an expert in some field
- the saying is famous
Is my understanding correct?