Glosses via Riverside Shakespeare (p. 1204):

berattle: cry down, satirize.

common stages: public theatres (the children played at the Blackfriars, a public theatre).

goose-quills: pens (of satirical playwrights).

Context via Shakespeare Online:

In 1600 a troupe of child actors (boy singers in the Queen’s chapel) began to act with great success at the Blackfriars private playhouse. In Jacke Drums Entertainement (written in 1600) occurs this passage:

[…]
Bra. Ju. ‘Tis a good gentle audience, and I hope the boies
Will come one day into the Court of requests.

The boys did come into request; indeed, they came into such request that the older playhouses suffered greatly. No less a person than Ben Jonson was engaged to write for the children; and the fashionable audiences that formerly patronized the public theaters, now turned to the private playhouse of Blackfriars.

For an extended commentary on this passage, see Roslyn L. Knutson’s “Falconer to the Little Eyases” (Shakespeare Quarterly, 1995).

This seemingly irrelevant question begins an extended digression into theater gossip, which has nothing to do with the Danish theater and everything to do with the London stage of Shakespeare’s own time.

This passage has occasioned much scholarly commentary for its topical references (see notes on child actors and the War of the Theatres below), which are often used in dating the play. In 1805, John Lord Chedworth confessed, “This passage, notwithstanding the pains bestowed on it by the commentators, I do not understand.” In the Riverside Shakespeare (2nd ed., p. 1187), Frank Kermode situates it within the context of an “extremely theatrical play” that:

does not pretend the stage is the little world. It reminds us that all this is occurring in a theatre, with the Ghost in the “cellarage” and the stage peopled by actors.

Though a probable late addition to the script (it appears only in the First Folio text of Hamlet, not the Second Quarto version), the passage suits the formal M.O. of the play, whose many digressions and delays reflect its hero’s. (In the same essay, Kermode points out that “Young Hamlet is not even mentioned until line 170–after nearly nine minutes' playing time.”)

For an extended commentary on this passage, see Roslyn L. Knutson’s “Falconer to the Little Eyases” (Shakespeare Quarterly, 1995).

Recall that Rosencrantz and Guildenstern haven’t been at Elsinore court lately, whereas Hamlet’s family situation (which is also a political situation) has effectively trapped him there, at home from university, for some time.

In a larger sense, Hamlet feels spiritually imprisoned in the claustrophobic atmosphere of Elsinore and the “rotten” state of Denmark. He has felt alienated from family, court, and country since his father’s death, doubly so since learning of his father’s murder. Rosencrantz and Guildenstern have little sense of any of this.

[Sent for us] for what purpose, my lord?

You tell me.

Hamlet knows why they’re there and soon tells them so–he is trying to get them to admit it for themselves.

conjure: implore.

consonancy: harmony, friendship.

withal: with.

Hamlet plays on the obligations arising from their childhood friendship in asking that Rosencrantz and Guildenstern tell him their true purpose.

lenten entertainment: stingy hospitality.

Rosencrantz defends himself against the suggestion that he reacted to Hamlet’s soaring oratory with a bawdy, sophomoric laugh about Hamlet’s sexuality. He suggests his smile was about the irony of what he knows about his friend–that his words and deeds do not match, that Hamlet might say no one can delight him but Rosencrantz knows that the men of the theater can and will.

If Rosencrantz did have bawdy humor in mind, it’s a good save.

In 3.2 Hamlet states that the purpose of playing is “to hold as ‘twere the mirror up to nature,” i.e. human nature–the very thing that Hamlet has just declared himself completely disconnected from.

Gary Oldman and Tim Roth in Tom Stoppard’s Rosencrantz and Guildenstern Are Dead (1990)

Polonius’s exit lines are usually played to suggest that he’s had enough–whether of Hamlet’s mockery or his bizarre behavior–and is making a quick escape.

https://youtu.be/J8C4gPU_hEU?t=1m19s

powerfully and potently: The Arden Shakespeare suggests Hamlet might be mocking Polonius’s repetitiveness.

honesty: “honest or honourable behaviour” (Arden); ties back to the earlier exchange.

Hamlet implies he very much agrees with the “rogue’s” mocking depiction of old men. The reason he gives for disputing it is deliberately absurd.

Polonius follows up on (and reiterates) the plan he’s explained to Claudius earlier.

We see clearly that Hamlet is baiting Polonius: in his “madness” he’s asking questions whose answers he knows perfectly well.