Why are the soldiers silent and motionless as the story begins?

This hanging is a solemn, formal occasion conducted in accordance with military protocol. Consequently the soldiers are under strict orders to remain at attention. The narrator explains the positions in which the soldiers are standing, except for the soldiers who are conducting the hanging itself.

Midway up the slope between the bridge and fort were the spectators--a single company of infantry in line, at "parade rest," the butts of their rifles on the ground, the...

This hanging is a solemn, formal occasion conducted in accordance with military protocol. Consequently the soldiers are under strict orders to remain at attention. The narrator explains the positions in which the soldiers are standing, except for the soldiers who are conducting the hanging itself.

Midway up the slope between the bridge and fort were the spectators--a single company of infantry in line, at "parade rest," the butts of their rifles on the ground, the barrels inclining slightly backward against the right shoulder, the hands crossed upon the stock. A lieutenant stood at the right of the line, the point of his sword upon the ground, his left hand resting upon his right. Excepting the group of four at the center of the bridge, not a man moved. The company faced the bridge, staring stonily, motionless. The sentinels, facing the banks of the stream, might have been statues to adorn the bridge. The captain stood with folded arms, silent, observing the work of his subordinates, but making no sign. 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.

"Parade rest" is not the same as "At ease." "Parade rest" is slightly different from strict "Attention" because the soldiers are not shouldering the rifles but resting them on the ground and holding them upright. These must be exceptionally long rifles because, as the narrator says, the barrels are inclining slightly against the right shoulder of each soldier. Evidently these are old-fashioned muzzle-loading rifles. A modern rifle would not come up much higher than a soldier's waist if the butt were resting on the ground. In modern "parade rest" the rifle would only need be held with one hand. The soldiers in either case are at attention and not permitted to move except upon command until they are dismissed. The lieutenant is standing at attention, and the two sentinels on the bridge are at attention.

The four men who are engaged in hanging the prisoner are a captain, a sergeant, and two private soldiers. They are the only soldiers who are moving because of the procedure of conducting the handing. The narrator explains that procedure in detail. It is ironic that Peyton Farquhar is being hanged from the very bridge which he intended to burn down. At the very end of the story his body will be hanging below the bridge and swinging gently back and forth. The body may be left there for a long time as a warning to any other Southerner who might have ideas about committing sabotage.

Ambrose Bierce uses the artistic device of contrast for effect. He emphasizes the silence and the rigidity of the participants because the scene which is described after Farquhar is allowed to fall with the noose around his neck is full of all sorts of violent action (at least in Farquhar's mind), as well as the sounds of rifle and cannon fire. All the soldiers suddenly come to life. There is intense activity on and near the bridge as well as in the turbulent water, where Farquhar imagines himself struggling to free his hands so that he can get to the surface and draw a deep breath. The strict formality of the opening scene is totally disrupted when the noose seems to break and the condemned man seems to have a chance to save his life.