If something can be said in one sentence, Polonius will say it in a paragraph; if something can be spoken in 30 seconds, Polonius will speak it in 5 minutes. The fact that he says "brevity is the soul of wit" is an ironic self-indictment--he's not brief, so he lacks wit. And did I say he's not brief?!
In this scene, particularly, we are struck by Polonius' artfulness in prolonging the simplest thing. He reminds the King and Queen that he's always been truthful, says he's always had their best interests at heart, tells them he knows something they don't, pauses for the messenger to deliver his message...and starts all over again. THEN he finally reveals his theory.
In adition to his "brevity" line, Polonius uses all these statements to tell the royal couple he's getting to the point: "I will be brief," "I will use no art," "I swear I use no art at all," "I will use no art," ""stay awhile, I will be faithful." In addition, he digresses into an I've-been-faithful-to-you-haven't-I speech as well as patting his own back that he had the foresight and wisdom to keep his daughter away from Hamlet. (In reality, of course, he was trying to protect his own reputation, not keep his daughter from potential heartache.)
Polonius is long-winded and rather foolish--which is what gets him killed, in the end. The King and Queen think he's long-winded, as does Laertes, his son, who rues the fact (in Act 2 scene 1) that he didn't make a speedier getaway and would now have to listen to another of his father's lengthy diatribes on life--before being scolded for being late to leave.
While all of these things are true, audiences do enjoy the irony of Polonius' words. He as a sympathetic character because he is a good-hearted guy with a little too much "gab"--and he doesn't realize it.
In this audience with King Claudius and Queen Gertrude, Polonious is wordy and is talking in circles. He is demonstrating the exact oppositite of "brevity." While the audience appreciates his wit and the scene is therefore funny for us, the king and queen are just anxious to hear what Polonius intends to reveal about what is wrong with Hamlet. When he says completely unnecessary things and reversals of those same lines, he seems "witty," but the wit is not appreicated as clearly indicated by the Queen's sarcastic response, "more matter with less art." Polonious is pleased to think that he has discovered the cause of Hamlet's changed disposition, thinking it is the result of the rejected love of Ophelia, when in truth, it is the knowledge of death of his father and his mother's hasty remarriage to Hamlet's uncle Claudius.
If there is one thing that Polonius isn't, it is brief or concise or any other word that means direct or to the point or not long-winded. Therefore, it is ironic that he is uttering this phrase. Even in this case, he takes line after line after line to basically say that Hamlet is mad, and that madness makes a man mad and that no one can deny that Hamlet is, in fact, mad.
Polonius is in many ways the fool of the play, though he is augmented in this role by Rosencrantz and Guildenstern, his foolishness is often highlighted in his interactions with Hamlet.