The Despair Event Horizon is a very vivid phrase that tells you what it means to some degree. An event horizon refers to the point of a black hole that once crossed, there is no chance of escape. Anything reaching this threshold is forever trapped in the gravity well. The Despair Event Horizon is quite similar.
This is the point that once crossed, there is no chance of recovery. This does not just refer to a character that is depressed or upset over something. The Despair Event Horizon is when something so traumatic happens that the victim of circumstance gives up completely.
There are a few ways of establishing this break. Most of them involve extremely gruesome circumstances. Remember this is breaking a person beyond repair. To a sadistic author such as myself, this is a very enjoyable experience. To a reader, this is the moment where they are likely to feel emotions towards the poor character.
One other trope that tends to come into play with this one is the "Hope Spot." This, in conjunction with the other trope, means that you give the poor character in question a moment where it looks like things will turn out just fine for them. Then the twist occurs. One of the most well-known examples of this happening in fiction is William Shakespeare's Romeo and Juliet. Both Romeo and Juliet are both driven across the line and then quickly are driven to suicide as a result.
However, there are other ways to employ this trope other than suicide. Think about it. In the mind of this character, his or her life is over. Everything is ruined; there is no hope. Now let's imagine how entertaining such a story development can be. A heroic character driven over the Despair Event Horizon might become a villain and begin causing horror upon the world that they have cursed. Another possibility is this despair being weaponized by another being. A God intentionally causing despair in mortals for a sinister purpose is a very common and effective means of creating a Cosmic Horror Story.
In my own works of fiction, I use this trope relentlessly. I am currently writing a story where basically the entire cast crosses the Despair Event Horizon to showcase just how cruel a world they live in. Characters who fall to such extreme despair make for Anti Heroes that are such a joy to write. Villains pushed to the brink no longer are held back by whatever morality they might have clung to.
This is a fantastic opportunity to make use of the Power of Hate and how it contrasts with the Power of Love. As a Deconstruction Dark Fantasy Writer, having hatred prevail in the face of love is one of my favorite past times. Writing downer endings to stories is my muse. For those looking to try something different and admittedly shocking for a story, why not give this trope a spin?
Just beware. Once you cross the Despair Event Horizon, there is no coming back...