Most Shavians are familiar with the above quotation. They are part of a longer speech by the Serpent in the first part of Back to Methuselah. I've chosen to expand the context a little bit, so we can grasp its full meaning.
THE SERPENT. The serpent never dies. Some day you shall see me come out of this beautiful skin, a new snake with a new and lovelier skin. That is birth.
EVE. I have seen that. It is wonderful.
THE SERPENT. If I can do that, what can I not do? I tell you I am very subtle. When you and Adam talk, I hear you say 'Why?' Always 'Why?' You see things; and you say 'Why?' But I dream things that never were; and I say 'Why not?' I made the word dead to describe my old skin that I cast when I am renewed. I call that renewal being born.
One of the reasons why we love Shaw quotations is their pithy wisdom. Another, is the fact that they have become objects of culture and they have been passed down both in popular discourse and in the words of other great thinkers and authors. I thought I could give you a glimpse of how iconic Shaw's words are sometimes by using this quotation as an example.
To begin with the best known paraphrase of these words, here's a video of John F. Kennedy's address to the Irish Gail (Parliament) in 1963. You can listen to Kennedy's rendering of the Shavian line at 16:55.
Five years later, during a speech at the University of Kansas, Robert F. Kennedy turned to Shaw again for inspiration. You'll hear Robert's slightly modified paraphrase of Shaw's quotation (although he acknowledges the source) at the 30:35 mark.
Still, I think my favourite instance of how this Shaw quotation can be used has to be the following fragment from Seinfeld (Season 3, Ep. 19, "The Limo"). I mean, 'why not?'