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!
In the first scene of William Shakespeare's tragedy Hamlet, the palace guards and officers are patrolling the castle's perimeter, as is their nightly duty. Francisco, a guard, and Bernardo, an officer, discuss the coldness of the night and prepare for the changing of the guard. Marcellus and Horatio arrive on the scene. Marcellus and Bernardo, both officers, are convinced that they have indeed seen a ghost about the premises. Horatio, however, is unconvinced, prompting Marcellus to comment:
"Horatio says 'tis but our fantasy,
And will not let belief take hold of him Touching this dreaded sight, twice seen of us . . ."
The ghost does appear in Horatio's presence, and the skeptic—and close friend and confidant of Hamlet—is now confronted with the reality of this mysterious apparition that bears a striking resemblance to the recently deceased king of Denmark. The appearance of the ghost prompts Marcellus to urge Horatio to address the ghost, as he, Horatio, is a scholar whose opinion is highly respected. Horatio, as with those before him, is struck by the ghost's physical similarity to the dead king, causing him to declare that it "harrows me with fear and wonder." Horatio, however, is abrupt and demanding of the ghost, resulting in the latter's departure. Anxious to hear Horatio's interpretation of this strange encounter, the scholar replies:
"In what particular thought to work I know not; But in the gross and scope of my opinion,
This bodes some strange eruption to our state."
Horatio interprets the ghost's appearances as an omen of tragedy to come. Whereas the tragedy of Hamlet will involve the young prince's efforts to avenge his father's death at the apparent hands of Claudius, the new king who has married Hamlet's mother, Gertrude, Horatio interprets the ghost's appearances as presaging terrible losses for the Kingdom of Denmark at the hands of the Kingdom of Norway, whose king was killed by Hamlet's now-dead father. In short, Horatio interprets the ghost's appearances as signifying ominous developments to come, but the real reason for the ghost's appearances, the audience will discover, is to compel Hamlet to avenge his father's death at the hands of Claudius.