Why at the end of Hamlet, Shakespeare refers at least four times to theater.Using words such as "stage" and "put on". At the end, Fortinbras orders...

The use of stage terms in the closing scene of
Shakespeare's Hamlet very directly reflects a line from another
Shakespeare play, but it's not Romeo and Juliet, it's As
You Like It:

readability="5">

All the world's a
stage...

Existence in
Hamlet is like being on a stage--everybody's
acting:

• Hamlet pretends to be
  mad.


• Claudius pretends to be honest and just, and
  pretends to care about Hamlet.  He toasts him in the closing scene, when he is really
  trying to get him to drink the poison in the
  cup.


• Polonius, Ophelia, Ros. and Guil. all spy on Hamlet,
  pretending to be his friend, or more, while actually trying to get information out of
  him. 


• The 1 Player pretends to care about
  Hecuba.


• Laertes pretends to forgive Hamlet, and treats
  him with respect, while he is in the process of trying to kill
  him.

All the world, existence, is a stage in
the play. 

Specifically, though, the theatrical terms you
mention all have literal meanings in the lines you cite.  For instance, "put on," when
used by Fortinbras, simply means that if Hamlet would have been "put on" the throne, he
would have proved most royal.

Figuratively, I suggest the
terms suggest the idea that life is a stage, and everyone acts like this or that;
everyone wears masks, plays roles.  This is much stronger in Hamlet
than is any suggestion of drawing attention to the fact that those putting on the play
are actors.  That is not central to this play, as it might be in Romeo and
Juliet.  Why?  Because there's no chorus in Hamlet. 
Shakespeare had matured and was past the use of a chorus by the time he wrote
Hamlet.  He does not go out of his way to draw attention to the
actors themselves, instead of the characters, in
Hamlet.