In Act 2 Scene 2 of Hamlet, what is the significance of the meeting between the royal couple and Rosencrantz and Guildenstern?

First, the scene provides a bridge from Hamlet's
declaration that he will "put an antic disposition on," in Act I, scene v, to this
scene.  Time has passed between these scenes.  We aren't told how much time, but it has
been enough time for Claudius and Gertrude to take note of Hamlet's "transformation" (as
Claudius calls it) and decide to send for Rosencrantz and Guildenstern (R & G). 
So, giving the audience the understanding that time has passed is one
significance.

Another, and more important, significance is
to expose the behind-the-scenes sort of machinations that Claudius is up to in the
play.  He has called Hamlet's friends to court to have them serve as sort of spies
against his own nephew.  This exposes how concerned Claudius is that he maintain his
seat as King, since, by true royal lineage, Hamlet should have been next in line for the
throne, not Claudius.  So, politically, Hamlet is a threat to Claudius. And, he seems to
waste no time, upon seeing Hamlet's odd behaviour, in deciding to get to the bottom of
Hamlet's actions.  He couches it in a concern for Hamlet's mental health, but his intent
seems pretty clear:

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. . .I entreat you
both

That, being of so young days brought up with
him,

. . .That you vouchsafe your rest here in our
court

Some little time, so by your
companies

To draw him on to pleasures and to
gather

. . .Whether aught to
us unknown afflicts him
thus

That, open'd, lies within
our
remedy.

These
final lines of his speech indicate that he wants them, like a pair of spies, to give him
information about Hamlet intentions.

Gertrude also supports
the idea that it's business and not friendship they are plotting with R & G when
she apparently offers them money for any information that bring the royal couple.  She
says:

. . .If
it will please you

To show us much gentry and good
will

As to expend your time with us
awhile

For the supply and profit of our
hope,

Your visitation shall receive such
thanks

As fits a king's
remembrance.

Gertrude,
like Claudius, is very good at couching this crass request in polite and oblique terms,
but the events of the rest of the play (when R & G are exposed as basically the
King's henchmen) bear it out.  The King and Queen are engaging the services of R
& G as spies in Act II, scene ii.  And the larger significance of this scene is
that it is quickly becoming apparent in this play that Hamlet can trust almost no one
(except Horatio) at this court.