Answer:
Polonius reads the love letter aloud to Gertrude and Claudius for two reasons:
He wants to show that he is a loyal subject, and that he's not trying to get his daughter together with Hamlet—Hamlet, being a royal, won't get much advantage from marrying the daughter of a mere noble. If he reveals the message to them directly, they'll know he is "a man faithful and honorable," as Claudius says.
He wants to get credit for being the one to know why Hamlet is mad. Once he reads the letter, he uses it to explain how Hamlet became mad: "he, repelled...Fell into a sadness, then into a fast...and, by declension / Into the madness wherein now he raves." Claudius was very eager to find out why Hamlet was mad, so being able to give an explanation makes Polonius look good.
His plans after reading the letter are to show the king directly that love is the cause of Hamlet's madness by taking the king to observe a conversation between Ophelia and Hamlet. As it turns out, Hamlet is very mean to Ophelia during this conversation, and the King concludes that he is not in love: "Love? His affections do not that way tend." So Polonius's plan doesn't really succeed.
