The prisoners are found near the end of the level in a Nazi camp. Going for this achievement/trophy will render Hedge-hopper unobtainable as after the prisoners are freed they will be unkillable thus the other achievements have to be done on other separate playthroughs.
Start by stealthily killing the Nazi executioner, as if the player takes too long, the executioner can kill the prisoners. If no one noticed, proceed to stealthily kill any lone patrolling Nazis. More than likely, Artyom will eventually be seen by Nazis. Take cover and dispatch any attacking Nazi. Stay away from the prisoners or else they might be caught in the crossfire. Eventually an armed trolley will appear. Take cover and wait for the trolley, though readers should note that the trolley must be destroyed - preferably with a sticky grenade. After the trolley is destroyed and all the Nazis are dead, Artyom should return to the prisoners, one of them should be standing. Talk to the leader, he will give thanks and the player will receive the achievement, and a moral point.