These lines are Horatio's, who is unafraid to confront the apparition. He seizes the opportunity to question the Ghost. First, he knows the Ghost, the former king, is distressed. A spirit in limbo cannot rest, and Horatio wants to know what he can do to relieve him.
Secondly, Horatio would like to know what kind of knowledge of the Ghost has about the movement of enemies, in particular Fortinbras, against Denmark.
Finally, Horatio thinks that perhaps the Ghost might want to disclose the location of some treasure. He wants to know where it is hidden and how to find it. It was thought that an unclear conscious could result in restless death.