Why does Shakespere show such a scene at the beginning of Hamlet?

I am going to assume that you are referring to Act 1 scene
1.  The reason that Shakespeare starts the play with several minor characters, rather
than the main protagonist, Hamlet, is because he wants to establish the background
information of the story that will follow, and because he wants to set the tone and mood
for the rest of the story.  You also have to remember that in Shakespeare's open-air
theatre, there would be no way to signal the start of the play except for having the
play itself begin.  The first several lines of the play are never VITAL, as the audience
would have just been settling into silence to hear the rest of the
play.

The content of the opening scene is immediately
intriguing -- after all, how many people have seen a ghost?  The audience is immediately
drawn in and starts to question why a ghost is appearing in the first place. Through the
guards conversation with Horatio, we learn that the ghost looks like the recently
deceased King Hamlet and we learn that this ghost doesn't seem to want to talk to any of
these men, so he must be signalling that he wants to speak to someone else, probably his
son, Hamlet.  Horatio also reveals all of the possible reasons a ghost may appear:
unfinished business, a warning, or to reveal hidden treasure.  This gets the audience
thinking about what could be THIS ghost's mission.

The
second purpose of the scene is reveal the background of a military threat being posed by
Fortinbras, Prince of Norway, whom we learn is planning an attack on Denmark to regain
land lost be his father several years earlier.  The guards ask about the increase in
military preparations, so we learn that the current king is doing his job in protecting
Denmark.  We also learn a little about Fortinbras's nature.  Horatio points out that he
is "hot" and of "unimproved mettle" suggesting that he is a bit of a hot head and very
determined.  He also reveals that Fortinbras's army is made up of mercenaries, not the
official army of Norway, again suggesting that Fortinbras is going to do whatever it
takes to achieve his goals.

The audience clearly
understands by the end of the first scene that there is potential internal  and external
problems for Denmark.  We are now drawn into these mysteries and are anxious to see how
everything resolves itself.