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At the end of Act III, Scene iii, Polonius says this:"Behind the arras I'll convey myself To hear the process; I'll warrant she'll tax him home: And, as you said, and wisely was it said, 'Tis meet that some more audience than a mother, Since nature makes them partial, should o'erhear The speech, of vantage."
From this, it seems that he is planning to listen in on their discussion because he feels like Gertrude will be "partial" (biased because he is her son) while talking to Hamlet and then sharing what they discuss. He thinks that if he hears it, he can listen and convey what is said more objectively to Claudius than Gertrude could.
A seperate answer to this question is that Polonius loves to meddle. Throughout the play, until he is killed, he is constantly trying to analyze Hamlet's behavior and get to the bottom of it for the sake of Claudius. He seems to love drama. He also volunteers himself to figure out what is bothering Hamlet in Act II, Scene ii. Polonius is also the one who suggests using Ophelia to figure out what is going on with Hamlet in Act III, Scene i, while he, Gertrude and Claudius withdraw and listen along.
It starts back in 1.3 when Polonius is talking to Ophelia. At the very end of the scene Polonius tells Ophelia that she is no longer "to give words or talk with the Lord Hamlet." Now go to 2.1. This is a scene where Polonius is instructing Reynaldo on how he is to spy on Laertes. Polonius thinks he is very clever with his spying acumen. After Reynaldo departs Ophelia appears in a tizzy. Hamlet has appeared at her closet acting crazy. Ophelia confirms to Polonius that she has been faithful to his word and denied access to Hamlet. Polonius immediately connects the two: Hamlet is mad from neglected love.
In the next scene Polonius tells the king that he has discovered the cause of Hamlet's madness. After the Ambassadors depart, Polonius clearly infatuated with holding center stage launches into bombastic circumlocution that convinces the king and queen to follow his plan. Polonius wants to arrange a meeting between Hamlet and Ophelia and see what happens.
In 3.1 the nunnery scene plays out but by the end the King is not convinced. He thinks Hamlet is dangerous and wants to ship him off to England. Polonius is firm in his belief that Hamlet's madness "sprung from neglected love." Polonius is undaunted and he gets Claudius to let him spy on Hamlet as his mother confronts him. She is to get Hamlet to "show his grief" and from Polonius' perspective this will prove his theory of Hamlet's madness.
After the play in 3.3 Polonius briefly encounters Claudius. He brings Claudius up to speed with his plan. He specifically tells Claudius that he is heading to Gertrude's closet where he will hide and spy on Gertrude's encounter with her son. And that's why Polonius spies on Hamlet and Gertrude in 3.4.
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