put someone out of his or her misery

Kill a wounded or suffering animal or person, as in When a horse breaks a leg, there is nothing to do but put it out of its misery . [Late 1700s]

End someone's feeling of suspense, as in Tell them who won the tournament; put them out of their misery . [c. 1920] Both usages employ put out of in the sense of “extricate” or “free from.”