Some scientific research has been carried out on human placentophagy (i.e. ingestion of the placenta). However, with the rapidly growing trend of placentaphagy, and placenta encapsulation in particular, more studies are being undertaken. See here for details of one of the first definitive studies on the effectiveness of placenta encapsulation that is currently underway by the UNLV. Placenta Benefits also has some related research information (see links below).
Giving birth is a time in your life when your hormones are under a major upheaval. Additionally you can potentially lose a larger than normal amount of blood which can leave you prone to iron deficiency. The placenta is believed to help progress the recovery from this process, and amongst other things, help avoid iron deficiency as well as its side effects (which include fatigue, depression and low mood). Despite the lack of major human studies, women who have had their placenta encapsulated swear by it and recommend it to others. See here for some of my clients' experiences of taking placenta capsules.