What is the significance of this quote to the plot of the play? "We are such stuff as dreams are made on, and our little life is rounded with a sleep."

This is one of Shakespeare's more famous quotes. Prospero's island represents the pastoral, where dreams and reality mix. The quote tells us that we ultimately derive meaning in our lives from our dreams and imaginations. As in his other Play "A Midsummer Nights Dream" or even "As you Like it", dreams and reality become one re-defining the lives of the characters. The true meaning of life isn't found in one's waking hours but rather in sleep.