Choose side ops for this as there are a few missions which focus on rescuing prisoners. Sneak in (helps if you interrogate an enemy soldier, they usually tell you where the captive is being held), get to the prisoner, use the Fulton recovery and they are counted as extracted. If the prisoner is held inside a building or tent, carry them outside so that the Fulton won't get stopped by the roof. Do this with 10 different prisoners and you should get your challenge pop up.
EDIT - if you rescue a prisoner by getting them to "da chopper", this also counts towards this challenge. I can confirm this unlocked for me on my 10th prisoner, during mission 10 - angel with broken wings.
Mission 10 - Angel With Broken Wings gives you the chance to rescue multiple prisoners, if you can get to them before they are executed.
This pops once you have completed the mission.
EDIT -as confirmed by Xtowers, there are at least 7 prisoners in mission 10. Advise that you rescue some prisoners, die and reload checkpoint. Each time you Fulton one out it counts.