The Shakespeare in this episode is carefully and thoughtfully integrated into the show's ending. The plot involves a pilot from World War I fleeing from a battle, leaving one of his comrades alone and in danger, by flying into a mysterious cloud. He lands in the 1950s, recognizes his cowardice, and flies back into a similar mysterious cloud to finish the battle.
In the clip below, the officers from the 1950s meet up with the pilot's comrade—who bore the nickname "Old Leadbottom" (but only among the few members of his squadron)—and say, in a very Shakespearean way, that they will recount to him the details of what's happened off stage.
At that point, the camera turns to the window to show us a mysterious cloud. The voice of Rod Serling closes the episode with the quotation that follows the clip—a quote that is very apropos of The Twilight Zone in general.
Dialogue from a play: Hamlet to Horatio: "There are more things in heaven and earth than are dreamt of in your philosophy." Dialogue from a play written long before men took to the sky. There are more things in heaven and earth and in the sky than perhaps can be dreamt of. And somewhere in between heaven, the sky, the earth lies the Twilight Zone.
Links: The Episode at Wikipedia.