Why, according to Aristotle’s theories about tragedy, would Sophocles make the Messenger’s description of Haemon’s suicide so graphic?
In Poetics 6, Aristotle says that the visual component of tragedy "is the least artistic, and connected least with the art of poetry" (S.H. Butcher translation). Aristotle repeats basically the same point in Poetics 14, but here stresses that merely by hearing a tragic tale, the playwright should be able to captivate his audience:
For the plot ought to be so constructed that, even without the aid of the eye, he who hears the tale told will thrill with horror and melt to pity at what takes place. This is the impression we should receive from hearing the story of the Oedipus. But to produce this effect by the mere spectacle is a less artistic method, and dependent on extraneous aids. (S.H. Butcher translation)
So, from Aristotle's perspective, it is more important for a playwright to be skilled with his words, rather than skilled at creating special visual effects (as in modern American movies). Therefore, instead of having Haemon kill himself on stage in front of the spectators, Sophocles has the Messenger describe Haemon's death.
This same practice of description is used throughout surviving Greek tragedy. Rarely, in Greek drama, does a person die on stage (Euripides' Alcestis contains one exception; considerable debate exists about whether Ajax's suicide in Sophocles' Ajax was visible to the audience).