As you play, you will unlock the cures for ailments that affect the citizens, I have found the most common to be fatigue.
This achievement is easily done after unlocking the fatigue cure and gathering all the materials at Pembroke Hospital. You can craft 10 of these and play a few nights until all the citizens have fatigue. I played for 3 in-game nights and several of the citizens across the city had it by the end of the 3rd night.
When you speak with a citizen, in the top right corner is a icon that says medical checkup (rb), press (rt) for details. You can press rb to ask them about needing medicine and then select the prompt on the right side to give them medicine. Do this 10 times with citizens in need to get the achievement.