As they took up arms and, in the phrase they commonly used to describe initiation into battle, went "to see the elephant," individual soldiers worried about their direct personal responsibility for the killing.
Compared to ideationally fractal animals like tigers, whose tigerness is evident all the way down to the fur, an elephant looks like an evolutionary garage sale of parts thrown together only to test the cognitive skills of the blind and parody the limited perspective and confident opinion with which we often greet the unfamiliar. I imagine that a live battlefield could also seem like a jumble of disconnected parts and restricted views, tied together only in retrospect by overwhelming terror and the abstractions of distant observers.
Still, "seeing the elephant" seems an odd way to describe going to war.
A more interesting (and likely) explanation is this joke, current in the decades before the Civil War, in which the elephant still represents the shock of the new, but now the impact is potentially fatal and your own heedless, thrill-seeking curiosity is ultimately to blame:
A New England tale tells of the farmer who had heard much of the unique appearance of the elephant, but had never seen one. He was told by a neighbor that the circus was expected in a nearby community and he swore he would attend. Setting out early on the following morning in his wagon, the farmer hoped to make the first performance. Upon reaching a crossroad several miles from home, where vision was obscured by a tall hedgerow, the farmer urged his horse into the intersection. At that same moment the circus train, led by the elephant, reached the crossroad from a different direction. The resulting collision smashed the wagon to splinters, killed the horse, and knocked the farmer unconscious. The circus train passed on as though nothing had happened. Awakening after several hours, the farmer surveyed the destruction and stated dryly, "Well, at least I've seen the elephant.