Upon being frightened, the Sim will jump in the air, fall back down, and their needs will drop. If their needs drop too low, they will check their pulse, and when they realize they don't have one, they will collapse and die.
Children can be frightened in this way, but can not die as a result but instead, they will wet themselves or faint. The ghost of a Sim who died of fright will have a pink tint. The ghost of a Sim who died of a vampire coffin scare will have a pale blue tint.
In The Sims 2, if a Sim's hunger bar reaches zero when being frightened by a ghost, they may instead die of starvation.