Why does the ghost want secrecy?
The ghost says "SWEAR" -- to Marcellus and Horatio, insisting they keep its existence secret.
Never to speak of this that you have seen.
...
Never to speak of this that you have heard.
...
(paraphrasing) not to give away Hamlet's game
I'm not sure why the ghost shakes the ground to get that promise from the others. (Incidentally, did you ever notice: they never actually swear?)
One more thing, I can't see proof in the text that the others hear the ghost. Horatio exclaims, "O day and night, but this is wondrous strange!" but he could be referring to Hamlet's "wild and whirling words."