To be amazed or shocked (usually negatively); to be exasperated or fed up with something so absurd or unreasonable.
彼のいい加減な言い訳には呆れて物が言えなかった。
I was so exasperated by his careless excuses that I couldn't say anything.
そんな値段を請求するなんて呆れるほどだ。
It's astonishing (in a bad way) that they would charge such a price.
同僚の無責任さに呆れてしまった。
I was appalled by my colleague's irresponsibility.