What is the dramatic importance of Act III, scene 1, with Rosencrantz and Guildenstern?
The short passages in Act 3.1 of Shakespeare's Hamlet that concern Ros. and Guil. deal with their reporting to Polonius and Gertrude what they were able to find out from Hamlet about why he is acting "mad." They were able to find out nothing.
The scene repeats the theme of acting and spying and pretending--that's what the two friends have been doing during their interactions with Hamlet. Pretending to be only his friends, they are acting on the king's behalf, trying to get information out of Hamlet. Together with Polonius and Ophelia, these two try to manipulate Hamlet and gleen information from him, while pretending to be other than what they are.
The scene also establishes Ros. and Guil. as instruments of futility (characterization). They may or may not be fools, but they are unsuccessful in their spying, and later they will certainly be treated like fools by Hamlet--payback for their trying to play him.
Hamlet's love for art is established, possibly, in his feeling "a kind of joy" at the news of the players' arrival. But, more importantly, we see in his eagerness for the king and queen to see the coming performance, that he is already thinking of his plot to catch Claudius's reaction to the play, and thereby prove Claudius's innocence or guilt. Thus, this scene furthers the plot.
Ironically, if the scene creates atmosphere, it is an atmosphere of futility and irony. Not only are Ros. and Guil. ineffectual, but so is Claudius. Though he is powerful and capable, he has met his match in Hamlet. Not only is Hamlet's plan to pretend to be mad working--the king is wasting an awfully lot of time trying to figure Hamlet's madness out--but Hamlet is setting Claudius up with the "play within the play." Claudius thinks he is playing a cat-and-mouse game with Hamlet, and he is: the only problem for Claudius is that Hamlet is the cat.
One interesting note that the previous and very thorough post did not go over is the rather incredible exchange between Hamlet and Ophelia. She has come to return to him "remembrances," signifying an end to their relationship and Hamlet is somewhat shocked, still not really aware of the fact that her father has decreed that they should no longer see each other.
He proceeds to take some of his anger out on her, but also in such a way that he urges her to protect herself in the future from men such as himself. For even though he is passing honest, he is quite the rogue, were he to admit all that has passed through his mind.
This scene is at the same time horribly painful and remarkably interesting, as Hamlet is still desperately trying to figure out what to do about his father's death, and also in this situation with Ophelia where he lacks enough information to make a really rational decision.