Marcellus wants Horatio to see the ghost because Horatio does not believe that there is one; also, Marcellus believes that if Horatio, a scholar, speaks to this specter, it will respond.
The play Hamlet opens with the changing of the guard on the ramparts of Elsinore Castle on a cold winter's night. Bernardo relieves Francisco, and they speak of "Not a mouse stirring" (1.1.10).
Then, Marcellus and Horatio enter. When asked if the guards if they have seen anyone, Bernardo replies that he has seen nothing. Marcellus tells them,
Horatio says ’tis but our fantasy
And will not let belief take hold of him
Touching this dreaded sight twice seen of us. (1.1.20-22)
Since Horatio does not believe in ghosts, Marcellus feels that if Horatio sees this ghost, it must be real. Hopefully, too, he can speak to it.
Just then the ghost enters, and Marcellus urges everyone to be quiet. Bernardo notices that it resembles King Hamlet
. He remarks upon this resemblance to Horatio. Horatio replies that it does, indeed, and it frightens him. Further, Marcellus urges Horatio to question the ghost. Horatio speaks to it:
What art thou that usurp'st this time of night
Together with that fair and warlike form
In which the majesty of buried Denmark
Did sometimes march? By heaven I charge thee, speak. (1.44-47)
But the ghost will not speak to him, and it walks away. Horatio exclaims that if he had not seen it, he would not have believed in the ghost. Marcellus asks if this ghost does not resemble the deceased king; Horatio declares that it certainly does. Later, Horatio expresses some concern about this ghost because such spirits are usually harbingers of misfortunes to come. He cites how they ran through the streets of Rome, speaking gibberish before Julius Caesar
Before departing, Horatio suggests that they tell Hamlet what they have seen and urge him to come because then the spirit will speak to him, his son.