The boss doesn't go into enrage mode by itself. Under the boss there is a pipe with a pedestal next to it, on this pedestal there is a cube, if you take the cube off the pedestal the boss will go into enrage mode and stay this way untill you return the cube.
Eventually the pipe will start to fill with red. As soon as it fills up completely players will see the effect on their characters and will be instantly killed.
Lifting the cube off the pedestal stops the pipe from filling up.
That's it lol

Once the Hanged Man goes into enrage mode he will be invulnerable to damage. He'll also start firing lasers about 90 degrees in front of him at random position.
If you're fighting a boss for the first time, you probably don't have cool defence modules, which means the laser will kill you in one hit.

Once approximately 2 minutes have passed and the pipe is almost full, one player runs to the pedestal and takes the cube, after which the player with the cube in his hands must run around from the shots flying at him and from electic puddles appearing on his way. If you are this player then you have to be careful and have maximum protection against electricity.

At this time the rest of the group should stand in front of the boss, yes, in front of the upcoming scary laser and shoot the boss in the face. In enrage mode the Hanged Man has only one vulnerable spot, which he shows a few seconds before firing the laser (core with yellow health bar). You can stand on the side, but then it's hard to aim.

It only takes two people with a bit more than 100k DPS in a group of 4 to smash a vulnerable part in 2-3 seconds.

Once the vulnerable part is smashed, the Hanged Man will be stunned before he can fire the laser, if there is a player with a cube in hand at the moment boss is stunned, then the red thing from the pipe will be reduced.

Smash the vulnerable part of the boss a couple of times to completely remove the red stuff from the pipe to avoid being insta killed.

After that, return the cube to its place. The boss will return to its normal state and become exposed to damage.

Good luck finding people who know what to do in this game with dead chat lol