How do the king and queen react to Polonius's hypothesis about what is wrong with Hamlet?

You need to look at act 2, scene 2 to find the answer to this question. Polonius comes to King Claudius and Queen Gertrude and tells them that he has found "the very cause of Hamlet's lunacy," which is very gratifying for them to hear; Claudius says he has "longed" to know it.
Gertrude is more skeptical, feeling that the cause is fairly obviously "his father's death and our o'erhasty marriage." However, Polonius is allowed to speak.
Polonius, in the event, does not really give a reason for Hamlet's "lunacy." He simply states several times that Hamlet is "mad" and then vows to find "the cause of this effect."
Gertrude is slightly unimpressed by Polonius's descriptive musings on this madness, demanding "more matter, with less art." Polonius then reveals he has letters written to Ophelia by Hamlet declaring his love. Evidently, Polonius wants the king and queen to infer, without his saying so, that it is Hamlet's love for Ophelia that is driving his madness. This is so that they will be more inclined to allow Hamlet to marry a woman who is not a princess, and indeed this is the conclusion drawn by Claudius and Gertrude.
Eventually, they agree that Polonius will "loose [Ophelia] to him" (send her to spend time with him) so that they can determine if this has a positive effect on Hamlet.