In this quote from Shakespeare's Hamlet, found in Act 1.2, Claudius is justifying his own actions. He is explaining to his subjects why he has done what he's done. Indirectly (from the writer's point of view, not the character's), he is also outlining one of the central issues of the early part of the play.
This scene is the audience's introduction to the court of Elsinore. When Claudius refers to King Hamlet's death, he's informing the audience that the king of Denmark has recently died, in addition to beginning his logical train of thought that justifies his hasty marriage to the dead king's wife, Gertrude. The memory is fresh, green, but people must continue to lead their lives. Thus, he married Gertrude. The kingdom must continue to remember King Hamlet, yet must also continue to live.
This logical argument then leads into the new king's questioning of young Hamlet. First, young Fortinbras has to be taken care of, as well as Laertes. But when those situations are quickly handled, Claudius moves to Hamlet. The logic is carried over: everyone else, though still respectfully remembering the deceased king, has recovered and moved on, why can't young Hamlet? Notice the logical chain: Life must go on, situations like Fortinbras must be handled; thus, Claudius married Gertrude. Look how well Claudius is handling things! Everyone else is doing it--why can't young Hamlet?
Here's what he says:
But now, my cousin Hamlet, and my son--....
How is it that the clouds still hang on you?
When everyone else has recovered, how is it that you are still upset?
And the play is on! Gertrude's hasty remarriage is a key issue in the play, and an event that Hamlet obsesses about throughout the play. Claudius attempts to justify it in this scene, and uses his logical argument to convince Hamlet that he, too, should go along with it.
His argument fails.