The Dark Realm is an endless world that is home to familiar enemies with increased enemy health, and rate of bullet fire. The Dark Realm can only be unlocked by standing near a Gibberer and not firing any bullets. (To change what button stops firing bullets, check the Controls setting in the main menu/pause menu.) After standing near a Gibberer for 5-7 seconds, you will be transported to the realm.
After that, Dark Realm portals will appear randomly throughout subsequent runs in any world. Once you're in a Dark Realm, you can head back through the same portal you entered to head back to your previous location. Players can farm the Dark Realm (assuming you're well equipped) for more organs and items that could help you in a normal or Desecrated run.