Bierce spends much of the opening of the story describing the scene of the execution. This includes the presence of sentries at either end of the bridge, the sergeant and soldiers who have charge of the prisoner, and the commanding officer. These functionaries are mandated by military regulations. Less clear is the rationale for the witnesses: the soldiers at parade rest arranged on the bank of the creek watching. Presumably, the witnesses are there because the prisoner is a civilian.
The silence of everyone involved is related to the solemnity of the event. Bierce writes,
Death is a dignitary who when he comes announced is to be received with formal manifestations of respect, even by those most familiar with him. In the code of military etiquette silence and fixity are forms of deference.
In other words, the silence of the men is a matter of respect, not for the prisoner, but for death itself, a kind of common enemy.
The silence is an important detail for Bierce. So much of the realism of the story turns on attention to sensory detail, but in a way, this absence of sound serves to foreground the terrifying reality of the moments right before death. There is also a kind of irony in it, as well. Farquhar's desire to gain glory by destroying the bridge, even though he is unable to join the regular army, has gotten him an unsought distinction: a military execution.