Lewis Carroll saw life as a dream, and as pointed out by Martin Gardner in his notes to Through The Looking-Glass in The Annotated Alice, Carroll returns to the question of life as a dream in the first paragraph of chapter 8, in the closing lines of the book, and in the last line of the book's terminal poem. Thus, both of the Alice books, as dream tales, are also veiled parables about the meaning of life.
Games "play" a prominent role in the books. In Alice In Wonderland, we have playing cards and croquet; in Through The Looking-Glass, chess; and in both books, there is plenty of the wordplay for which Carroll is justly famous. Carroll himself was very fond of games--one reason he enjoyed spending time with children. But just as nonsense contains a deeper level of meaning, so do games. It has been said that chess, for example, with its black and white pattern of squares, was created as a reminder of the vast expanse of the field of existence, and how to navigate it. In essence, the chessboard or checkerboard symbolizes the polarities, the positive and negative forces, the yin and yang that must be kept in balance if the game of life is to be played well. The theme of polarity shows up in many and various ways in TTLG, starting with the black and white kittens in the first chapter. As Alice puts it when she first beholds the chessboard playing field: "It's a great, huge game of chess that's being played--all over the world--if this is the world at all, you know."
From the scientific standpoint, polarity is electromagnetic energy vibrating between two poles, which comprise a unity. Thought is also energy, vibrating at frequencies that cannot be measured with our current technology, and between two polarities. This gives rise to duality: the tendency for human thought to polarize to one of two extremes, to separate and compartmentalize. Linearity is perhaps the primary way we do this, in our perception of past, present and future time, which too often takes us out of the present moment. The White Queen's rule of "Jam yesterday, jam tomorrow, but never jam today," is just one example. Other manifestations of duality that show up in both Alice books, often exaggerated and/or parodied, are: fearfulness, confusion about identity, hierarchy, loneliness, wanting what is distant or unattainable, self-deprecation, and black and white thinking. Martin Gardner writes in his notes in The Annotated Alice: "In a sense, nonsense itself is sanity-insanity inversion. The ordinary world is turned upside down and backward; it becomes a world in which things go every way except the way they are supposed to." I would say it's a polarity parody, containing as many layers of symbolic meaning as the chessboard itself.
I'll close this with a quote from The Seth Material by Jane Roberts that for me seems to sum up Lewis Carroll's take on "life as a dream":
Humanity dreams the same dream at once, and you have your mass world.
The whole construction is like an educational play in which you are the producers as well as the actors.
There is a play within a play within a play.
There is no end to the "within" of things.
The dreamer dreams, and the dreamers within the dreams dream.
But the dreams are not meaningless, and the actions within them are significant.
The whole self is the observer and also a participator in the roles.