Weighted blankets have traditionally been used as part of occupational therapy for children who experience sensory disorders, stress and anxiety, often related to autism. Adult psychiatrists have more recently found weighted blankets to be a powerful tool for helping adults who are anxious, upset and possibly about to loose control.
The below excerpts cite independent professionals Temple Grandin, Ph.D. and Karen Moore, OTR/L.
How can something so simple as sleeping with weighted blankets be a solution to stress, anxiety, insomnia and more? Messed up sleep can create a long list of secondary issues that can quickly become primary concerns if insomnia or other disturbances continue untreated. Lack of sleep, whether it’s medically related or anxiety-driven, can throw off your normal functioning during the day. Concentration becomes difficult, productivity at work or school begins to suffer, irritability can have you lashing out at family and friends, and you also become at risk for serious health issues like heart attacks.
Deep pressure touch stimulation (or DPTS) is a type of therapy that almost anyone can benefit from. Similar to getting a massage, pressure exerted over the body has physical and psychological advantages. According to Temple Grandin, Ph.D., “Deep touch pressure is the type of surface pressure that is exerted in most types of firm touching, holding, stroking, petting of animals, or swaddling. [sic] Occupational therapists have observed that a very light touch alerts the nervous system, but deep pressure is relaxing and calming.” 
How does it work?
A weighted blanket molds to your body like a warm hug. The pressure also helps relax the nervous system. It’s a totally safe and effective non-drug therapy for sleep and relaxation naturally. Psychiatric, trauma, geriatric, and pediatric hospital units use weighted blankets to calm a patient’s anxiety and promote deep, restful sleep. In a similar way to swaddling comforting an infant, the weight and pressure on an adult provides comfort and relief.
When pressure is gently applied to the body, it encourages serotonin production, which lifts your mood. When serotonin naturally converts to melatonin, your body takes the cue to rest. 
More Than Just Insomnia And Anxiety!
Depression, anxiety, aggression, OCD, PTSD, and bi-polar disorder have all been linked to low serotonin levels in the brain, which weighted blankets are reported to assist with. In addition, people battling with depression, mania, anxiety, trauma, and paranoia, or undergoing detoxification have reported relief from symptoms.