Oh, my offence is rank. It smells to heaven.
It hath the primal eldest curse upon ’t,
A brother’s murder.

In a soliloquy—the only soliloquy in the play not spoken by Hamlet—Claudius admits murdering his brother, and he describes his guilt in the language of decay. His crime smells bad, like something rotting. Throughout Hamlet, moral faults are described in the language of rot, decay and disease. The Ghost’s “foul crimes” must be “purged away” remark in Act 1, Scene 5 references the Renaissance medical procedure of purging and in Act 3, Scene 4, Gertrude will see “black and grievèd spots” on her soul. And most famously, in Act 1, Scene 4, Marcellus the watchman told us that something was “rotten” in the state of Denmark. Like Claudius’s reference to rank smells here, these lines create a sense that the decay and sickness that infect everything in Hamlet has a spiritual dimension. The offending characters are not just doomed to die but doomed to damnation.

You can read more about this quote in Quotes by Character: Claudius (the third quote).

This physic but prolongs thy sickly days

Hamlet utters this line at the end of a crucial moment in Act 3, Scene 3 in which—after finding Claudius alone and deep in prayer—he passes up the opportunity to kill him because he worries that Claudius might be forgiven by God if he dies while praying. Hamlet vows to kill him later in an unsanctified moment (when Claudius is either drunk, angry, or lustful, he says) and then concludes with this quote. He is essentially saying that, while the king is trying to cure himself with the medicine (“physic”) of prayer, all he is really doing is keeping his sickly self alive slightly longer.

We have no reason to doubt Hamlet’s sincere belief in the religious grounds he evokes for not killing Claudius here when he has a clear opportunity to do so. Indeed, we saw Hamlet decide against suicide based on his religious beliefs in Act 3, Scene 1, so we can assume he is being sincere here as well. Regardless of his reasoning and his sincerity, however, the fact remains that Hamlet has yet again allowed uncertainty and delay to prevail over fast and decisive action in this scene.

My words fly up; my thoughts remain below.
Words without thoughts never to heaven go.

Moral and religious questions play a strong role in Hamlet, especially in Act 3, Scene 3, where both Hamlet and Claudius wrestle over quandaries in that sphere. With these lines that close out the scene, Claudius reaches the conclusion (which is firmly based in religious doctrine) that it is meaningless for him to repent for a sin (killing his brother) that he continues to benefit from and when he has no intention of giving up those benefits. Read more about this quote in Quotes by Character: Claudius (the fourth quote).