The play begins with a question and leaves us with many questions before we finish. In fact, Shakespeare poses many more questions than he answers with this play.
As with many of the first few lines of a new scene, Shakespeare must set the scene, let us know that it is night, that it is dark and hard to see, that it is cold, that these are watchmen who are fearful about something. There are many other lyrical descriptive passages which set the scene.
This is the first time the term "the Dane" has been used to refer to the King of Denmark, a title Claudius claims in the second scene of Act 1 [make link], and Hamlet seems to take back in Act 5, scene 1[make link]
Horatio is being humorous here and making fun of the guards who have called him to the battlements to see if he can speak to a ghost. Horatio does not quite believe in ghosts at this point, though because he is a scholar, he knows the proper way to speak with ghosts. Horatio is saying his body is there with them, and his spirit, if not he belief in ghosts yet. In other words his scepticism is there in place of his belief.
Often Shakespeare makes the metaphor fit the character. Here are guards standing watch, waiting for Fortenbras or someone to attack or assail them in their well-fortified castle, suggesting that they must use forceful persuasion to convince Horatio that they have indeed seen the ghost.
Although you see stage directions in this version of the play, there was nothing but the text itself to tell the editors when someone (or in this case, something, the ghost) is to enter. Of course, this leaves some room for ambiguity. Has the ghost been there and someone just now noticed? Or has the ghost just entered? And why would one of the characters inform the other characters about spotting the ghost? Obviously, here, since it is dark and since their purpose for being there is to see and try to speak to the ghost, whatever Marcellus is saying is not as important and Bernardo can interrupt him to point out the ghost.
In the Renaissance Humanist belief structure, everything was ordered and every possible question was answered by religion. Students, like Hamlet and Horatio, and Polonius before them, learned the lists of how to speak to a ghost, the three reasons a ghost would come back to earth, how one turns from a lover into a madman, and many more lists, so everytime you hear a character reciting what sounds like a memorized list, what is expected may be coming from this belief structure.
Apparently seeing is believing for Horatio, for he is no longer skeptical about ghosts.
The idea that the universe gives signs to the State of Denmark in this case, through storms or dreams or in this case a strange visitation by a ghost, is part of the Renaissance Christian Humanist belief that there is a plan that God has for all mankind, and that Nature reflects what happens with the important characters, like kings or caesars.