In the quote "My lord, as I was sewing in my closet, Lord Hamlet, with his doublet all unbrac’d No hat upon his head, his stockings foul’d Ungarter’d and down-gyved to his ankle, Pale as his shirt, his knees knocking each other, And if he had been loosed out of hell To speaks of horrors, he comes before me," who is speaking? Who are they speaking to? What is the context of the quote? What is the significance of the speech?

This incident occurs in act 2, scene 1. Ophelia is speaking. She informs her father, Polonius, that she was frightened by Hamlet when he suddenly entered her room in a disheveled condition. She expresses concern about Hamlet's seemingly strange behavior. The manner in which he looked at her was very disconcerting, and he sighed in a piteous and profound way. He then left the room while continuously staring at her, without seeing where he was going.
Ophelia's speech is significant because it reflects Hamlet's decision to "put an antic disposition on." He had told Horatio at the end of act 1, scene 5, that he would adopt this attitude so that others may think him mad. The reason for this would be to confuse all and sundry about his true mental disposition so that they would not realize that he is plotting to avenge his father's murder at the hands of his uncle, Claudius, who has married his mother and is now king.
Hamlet learned about this when confronted by his father's ghost in act 1, scene 5. The spirit told him Claudius had poured poison into his ear while he was asleep in the garden. He beseeched Hamlet to avenge his murder but that he should not harm his mother.
Polonius believes that Hamlet's actions are a symptom of his infatuation with Ophelia, and he tells her that they should inform Claudius about Hamlet's strange behavior. This decision would lead to their spying on Hamlet and ultimately result in Polonius's untimely death.

Ophelia is speaking in this passage, which occurs in Act II, scene 1. She is speaking to her father, Polonius.
Although it is unclear to Ophelia what is going on or why Hamlet is so upset when he bursts in on her, we as an audience know the context: Hamlet has just seen and spoken to the ghost of his dead father. The ghost has informed Hamlet that he was murdered by Hamlet's uncle, Claudius. That is why Hamlet comes in with his clothes loosened and in a disarray
However, when Polonius asks her if Hamlet is "mad" for her love, Ophelia says she fears that it might be so. She also tells Polonius that Hamlet held her wrist and stared at her for a long time.
This speech is significant because, based in it, Polonius misinterprets Hamlet's actions as lovesickness for Ophelia. Polonius will go on to inform Claudius, who is worried, that Hamlet is behaving oddly because he is in love with Ophelia.