The mood is rather tense on that cold winter's night on the battlements of Elsinore. The Danes are expecting Fortinbras to invade at any moment, and so naturally everyone's on edge. But the general air of anxiety that pervades the opening scene of Hamlet is as much related to what's happening on the domestic front as it is to the threat of imminent invasion. Claudius's recent acquisition of power has created a dark, unpleasant atmosphere in the kingdom. Something is indeed rotten in the state of Denmark and the mood of the opening scene reflects that.
Under the circumstances, it's hardly surprising that the watchmen claim to have seen a frightening apparition stalking the battlements. People who claim to see ghosts are often in a state of anxiety; the spirits that they claim to see could well be interpreted as manifestations of inner turmoil. And there's certainly plenty of that among the watchmen at the start of the play as there will be later on in the tormented soul of Hamlet himself.