First of all, it is worthy to note that in the Elizabethan Age, there was a strong belief in the supernatural; ghosts, for instance, were thought to have died of horrific circumstances, so they were to be feared. (Interestingly, in one of the paradoxes of this play, Hamlet's ghost comes from Purgatory where he is confined in fire until his "foul crimes" are "burnt and purged away." But, later in the drama, Hamlet resists killing Claudius while he prays because by murdering him in prayer, Claudius will have his sins purged. This doctrine pertaining to King Hamlet is a Roman Catholic one while Hamlet's belief in a "leap to grace" is a Protestant one.)
Clearly, there is a conditioned fear in Hamlet's friends when they see the ghost in Act I, Scene 1, especially as he appears to be King Hamlet dressed in his war armor which he wore when he defeated the King of Norway, Fortinbras. Horatio is concerned about this resemblance because King Claudius has returned to "this Fortinbras," the son, the lands that were taken by King Hamlet. In fact, Horatio notes, they are on guard to watch for an invasion from Norway. Horatio, then, suggests that he and the others tell Prince Hamlet about what they have witnessed.
***In Act I, Scene 4, the ghost reappears before Hamlet. After the ghost appears, he "beckons" the prince to follow him; Hamlet decides to do so because it will not speak, but Horatio begs him not to go. He cautions Hamlet that the ghost could do him some harm:
What if it tempt you toward the flood, my lord,
Or, to the dreadful summit of the cliff
That beetles o'er his base into the sea,
And there assumes some other horrible form
Which might deprive your sovereignty of reason
And draw you into madness? Think of it. (1.4.53-58)
Horatio is worried that Hamlet may drown in the ocean, he may fall from a cliff, or that he may be so terrified by a "horrible" shape that the ghost may assume that Hamlet loses his mind from the experience.