Marcellus and Bernardo want Horatio to speak to the ghost because he is a scholar.
Marcellus tells Horatio to speak to the ghost, even though he says it is nothing but their “fantasy” (Act 1, Scene 1), because they think that he will get further with it than they did. After all, it looks like the king, so it might be a royal ghost, so it will take a scholar to talk to it. Horatio will know what to say.
In the same figure, like the King that's dead.
Thou art a scholar; speak to it, Horatio.
Looks it not like the King? Mark it, Horatio. (Act 1, Scene 1)
Horatio is a little stunned, saying that the ghost fills him with “fear and wonder.” When he asks it if it was the King of Denmark, Marcellus tells him he has offended it and Bernardo says it “stalks away.” He tells Horatio to admit it is not something of fantasy.
Before my God, I might not this believe
Without the sensible and true avouch
Of mine own eyes. (Act 1, Scene 1)
They talk some more, and the ghost comes back, and Horatio tries again to talk to it. He begs, in fact, for it to speak to him, and even asks Marcellus to stop it. Marcellus asks how he is supposed to do that, by striking it? It leaves, and Marcellus says it is wrong to be violent to something so “majestical.”
Horatio is Hamlet's friend, so if the ghost is going to talk to anyone, it makes sense that it would be him. The others are basically just guards. However, the one the ghost really wants to talk to is Hamlet. The assumption that a scholar is needed here is a false one. What they really need is a prince.
Hamlet's ghost is wandering because he can get no rest until he gets revenge on his wife and brother, Claudius. Gertrude married Claudius after he killed Hamlet (Senior), and he will not be able to rest until he gets his son to kill the new king. This is the real reason he keeps appearing and disappearing. He wants his son to come, so he can tell him this. By the end of the play, Claudius will indeed be dead, but so will Gertrude and almost everyone else. Revenge is a messy game.