Why does Polonius say that he will spy on Hamlet's conversation with his mother?

Polonius is attempting to help King Claudius, Hamlet's uncle and step-father, ascertain the cause of Hamlet's strange behavior.  Though Hamlet's behavior changed with the death of his father and the very quick remarriage of his mother to his uncle, his oddness has deepened in ways that seem inexplicable.  They worry that he is going "mad."

To get to the bottom of this, Polonius and Claudius stage a scene where they can observe an interaction between...

Polonius is attempting to help King Claudius, Hamlet's uncle and step-father, ascertain the cause of Hamlet's strange behavior.  Though Hamlet's behavior changed with the death of his father and the very quick remarriage of his mother to his uncle, his oddness has deepened in ways that seem inexplicable.  They worry that he is going "mad."

To get to the bottom of this, Polonius and Claudius stage a scene where they can observe an interaction between Hamlet and Polonius's daughter, Ophelia, who used to be in a relationship with Hamlet until Polonius made her break it off.  Polonius believes, having heard the two speak together, that "The origin and commencement of his grief / Sprung from neglected love" (3.1.191-192).  In an effort to get Hamlet to reveal the cause of his grief, Polonius suggests that they "Let his queen-mother all alone entreat him" to open up to her (3.1.196).  He says, 

Let her be round with him;
And I'll be placed, so please you, in the ear,
Of all their conference.  (3.1.177-179)

Polonius volunteers to hide himself in the room so that he can hear everything that Hamlet says and report it faithfully back to Claudius.  He elaborates on this a bit more, later, when he says,

'Tis meet that some more audience than a mother,
Since nature makes them partial, should o'erhear
The speech of vantage. (3.3.34-36)

In other words, since mothers are partial to their children, Gertrude might be naturally inclined to report the conversation to her husband somewhat less faithfully than one who sees Hamlet more objectively.  Polonius doesn't imply that she would have any intention to deceive, but, rather, that she would inherently want to protect her son and keep his confidence.