To be very cooperative and submissive; to obey and believe someone without question.

This expression from the 20th century, describes what a tame or trusting animal will do if you treat it right. The person who created this idiom applied the same idea to human beings who trust fully and obey without question. People don't actually eat out of anyone's hand, but they do behave like obedient animals sometimes.

(From the "Dictionary of Idioms," Marvin Terban, Scholastic Inc., publisher, 1996)