From Blood on the Clocktower Wiki
The Doomsayer speeds up large games that are going too slowly.
"If 4 or more players live, each living player may publicly choose (once per game) that a player of their own alignment dies."
The Doomsayer allows players to sacrifice their allies in order to gain information, which shortens the game. Only living players may use the Doomsayer power, and may do so only once per game. It is their responsibility to remember to not use it again.
If a player says something like "I use the Doomsayer Power", then it is your choice as a Storyteller which player to kill, but you must kill a living ally of the player that used the Doomsayer power. So, if a good player uses the power, then a good player dies. If an evil player uses the power, then an evil player dies.
Obviously, you won't want to kill the Demon this way, unless the game can continue afterwards for some reason, such as a Scarlet Woman being in play.
Once 3 players are left alive, the Doomsayer ability may no longer be used.