Once Horatio has actually seen the ghost with his own eyes he believes that it is a real ghost and not just a figment of the guards imaginations, or to use Bernardo's word, a "fantasy." Horatio has been brought out to the this place because he is a school friend of Hamlet's and a scholar, not just a common guard. The three guards are hoping to confirm their sight and perhaps even get some intelligent guidance on how to next proceed in regards to this ghost. Clearly, the ghost doesn't want to talk to any of them, but his presisent visits suggest that they need to do something. I would guess that none of the men is real anxious to tell Hamlet that the ghost of his dead father is wandering around, but at least if Horatio, his trusted friend, tells him the news, it will be bettered received and trusted.
As Horatio explains why he trusts his vision, he explains that it looks like King Hamlet and it is dressed like King Hamlet once was. Horatio's conclusion is that this ghost "bodes some strange eruption to our state." He is well aware of "ghost-lore." He knows that ghosts only come when there is unfinished business or a warning to be conveyed -- there is no good/positive reason for a ghost's appearance!