Polonius feels it is necessary to inform Claudius that Hamlet is showing signs of madness; however, as a pathetic toady to the king (as he is often portrayed), it is very difficult to get the words out since he is risking angering the king.
It adds to Polonius' characterization as a man who is insecure in his position, in the court and in his family. (Or... it can lend itself to a characterization as an arrogant man who is overly secure in his position-- depends on the director.) One can draw parallels between this scene and the one where Laertes is leaving for University. "This above all: to thine own self be true" follows a lengthy speech offering mountains of advice to his son. In both speeches, Polonius speaks in a curcuitous manner: perhaps to speak cautiously, perhaps to hear himself speak.