to be sorry; to beg someone's pardon — a polite/formal phrase used to apologize or show humble regret.
お待たせして恐れ入ります、ただいまご案内します。
I'm sorry to have kept you waiting; I'll show you now.
恐れ入りますが、こちらにご記入ください。
Excuse me, but please fill this out here.
to be much obliged; to be grateful — to express gratitude with humility.
ご配慮いただき恐れ入ります。本当に助かりました。
I'm very grateful for your consideration. It really helped me.
資料を送っていただき、恐れ入ります。
Thank you very much for sending the materials.
to be amazed; to be greatly impressed — to express admiration or being impressed by someone's skill or achievement.
その速さには恐れ入ったよ。まさか一晩で仕上げるとは。
I'm impressed by that speed. I didn't expect it to be finished in one night.
彼女の演奏には毎回恐れ入ってしまう。
I'm amazed by her performance every time.
to be dumbfounded; to be confounded; to be defeated or fed up — to express being beaten, astonished, or at a loss.
君の論理には恐れ入った。反論の余地がない。
I'm beaten by your logic. There's no room to argue.
あの仕事量には恐れ入るしかない。
I can only be astonished at that amount of work.