Why have Marcellus and Bernardo asked Horatio to join them in their watch guard atop castle of Elsinore?

Bho ... NANI?!

In Act I Scene I Marcellus and Bernardo have asked Horatio to come and see with his own eyes the apparition that they have been seeing for the last couple of nights. They had told Horatio about this, but he didn't believe them, so he joins them to check whether their story is real or not. The quote that you will find below backs up this idea:

"Horatio says 'tis but our fantasy

And will not let belief take hold of him ( ...)

Therefore I have entreated him along

With us to watch the minutes of this night,

That if again this apparition come

He may approve our eyes and speak to it" Act I Scene I Line 24-25

To these words uttered by Marcellus, Horatio replies:

"Tush, tush, 'twill not appear." (=Nonsense, it will not appear) Act I Scene I Line 26

So it can be clearly seen that the characters brought Horatio to prove that they were telling the truth.