There are many causes for a villager to become weakened. They can be weakened by starvation, disease, on-the-job injuries, old age, and Island Events, to name a few. Some occupations are more hazardous than others (especially research and building), so moving them to a different job can help. Healers can only help a villager who is sick (check their details screen to be sure). Placing a healer on a weakened villager (who is not sick) will initiate an embrace.

The only way to get weakened villagers completely healthy again is to eliminate the cause of their deterioration (get more food, heal them, etc.) and allow them to rest and eat. Taking them to the food supply and repeatedly making them eat (at least 8-10 times) will improve their health, as will certain potions, but they will get better on their own as long as they don't suffer further damage to their health. The most important thing is to take care of whatever caused the weakness.