The bear's cage of a tornado is a slang term. A storm chaser will often use slang to describe a particular part of a storm. During the development of severe weather, a thunderstorm will often have a region of heavy precipitation and rotation which could mean the development of a tornado
. To say that you are in the bear's cage
means the area is dangerous and a tornado could (or already did) develop. In essence, this area is like a cage of rain surrounding the updraft of a tornado.
This is especially true for an HP supercell. These high precipitation supercell thunderstorms often includes damaging hail, flooding, and lightning. There can be such a wall of water that a tornado behind the curtain of rain can be hard to get a visual on. Even experienced and expert storm chasers will avoid this area.