The answer is quite simple, to “approve and avouch for the coming of the apparition with the gross and scope of his intellectual opinions”.
Indeed, Horatio is a scholar from the University of Wittenberg. The guards are uncultured and possibly illiterate they do not trust their judgment and defer to Horatio to ascertain the exact nature of the ghost.
In the Middle Ages, ghosts were regarded with suspicion. It was considered that they might be either a ruse from pranksters, a demon in disguise, a hallucination and if all of the former had been disproved, the actual wandering, tormented soul of a deceased human being.