It is clear how a word meaning “to rest” becomes quiet — it’s hard to rest when there are jackhammers outside, as there coincidentally are right now! — but how does a word meaning “to rest” become “to remain”?
The answer has to do with the notion of, what remains after everything else leaves. The food is sizzling hot — but it’s the quiet, sad pieces just sitting there, that no one wants, that remain. There’s a lot of noise and ruckus — and when all is said and done, only silence remains. Life is tale, full of sound and fury… and nothing remains (the Bard almost wrote!).
We can clearly see the qu-d of quedar map to the qu-t of quiet.