<em>Through the Looking-Glass </em>is a novel written by Lewis Carroll as the sequel to <em>Alice's Adventures in Wonderland.</em>
In the given scene, the Red Queen reveals to Alice that the entire countryside is laid out in squares, like a huge chessboard, and offers to make Alice a queen if she can move all the way to the eighth rank/row in a chess match.
The symbolic meaning that can be drawn from the given excerpt is that Alice is trying to grow up too quickly. It seems like she wants to become a queen before it's time, before she has passed the proper examination.
