she grabbed peter by the arm. 問什麽這裡的by the arm 是解作peter的手臂？

grab X by the Y = 拿住X的Y
catch X by the Y = 抓住X的Y

就醬簡易!

Eeny, meeny, miny, moe, catch a tiger by the toe.

She took up and held Peter's arm.
She made a grab at Peter's arm.
She grabbed at Peter's arm.