1. To become numb to one's emotions or outside sensations as a result some physical or psychological stimulus or trauma. As a way of dealing with my depression and anxiety, I spent several years in college just numbing out with booze and video games. A large portion of the population is just numbing out when it comes to voting because they feel like the system is inherently broken.
2. To cause someone to become numb to their emotions or outside sensations as a result some physical or psychological stimulus or trauma. In this usage, a noun or pronoun is used between "numb" and "out." The loss of my daughter in the accident completely numbed me out for the better part of 10 years. My doctor prescribed this medication to help me cope with my anxiety, but I find that it's been really numbing me out, so I might ask her about going off it for a while.