척하다 is putting on an act: 자는 척 하면서 'pretending to be asleep', 알아 듣는 척 하면 'if you pretend to understand', 못하는 척 하는 거 아니에요? 'aren't you pretending to be bad at it?'. Note one line uses 아는 척하다 in its other, non-deceptive sense — greeting or acknowledging someone you know: 길에서 만나면 꼭 서로 아는 척하면서 'we always say hi when we run into each other'. The bookish equivalent of 척하다 is 체하다; the meaning is the same.