The Age Line Of The Book
This can be a subject of a long and detailed research, but, in short, the books go in the line of growing up from a little girl to a women by the persons Alice meets. Here some examples to this process:

  1. Tweedledum and Tweedledee are two little kids that fight over a rattle.
  2. Humpty Dumpty is a teenager so proud of his cravat and trust the King (father ?) to help him in case of trouble (fall).
  3. The Lion and The Unicorn are two young men trying to find their place as males (are they fighting over Alice).
  4. The White Knight is a man in it's full power.
  5. At d8 Alice becomes a Queen = Woman.

This line of thinking can be carried forward much longer and deeper, but I made my point and leave the rest to better Carroll experts.

The Chess Game
I think the book is based on a game Carroll had with Alice. The moves are not simply mad. To my feeling, it's a game between a little girl and an adult that tries not simply winning the game but more important, to show her and teach her a little about the basics of it.

Why Does The White Moves Several Times In A Row ?
The fact that one side is doing several moves in a row brings to mind another great English Mathematician: Allen Turing. The story goes that Turing used to play "running chess" with his friend: Once you do a move, you go run around the house, if the other didn't move until you return, you play again...

Why Are The Chessmen Red & White ?
Carroll could pick any color, and one simple answer is that it's a natural follow- up to the White/Red roses in Wonderland, but I think Carroll had other roses in mind:
I think these colors were used as a reference to the War of the Roses in England 1455-1485. The similarity is interesting, but due to scope limitation I'll have only to hint. In this war, two families where fighting on the English crown: the Yorkists who's mark was white rose and the Lancastrians who's sign was red rose. During this war over the crown, the women of the families played important roles. At the end a new winner came: the Tudors that Queen Victoria is of that family. The Tudors claim to continue both roses via Henry VII wife & mother just like Alice was crowned by both red and white queen together.

The Chessboard
I don't know why but it can't be just luck that all the battles and the violent conflicts are happening on non- white (red ? black?) squares:

  1. Tweedledum and Tweedledee are going to fight when Alice on d4.
  2. The Lion and The Unicorn are fighting when Alice on d6.
  3. The Red & White knights are fighting at e7.
  4. When Alice reaches d8, she is going out of it to kill (eat) the red queen.

Why is it so ? good question, the only idea I have now is that white is a symbol of peace and red, on the other hand is the color of blood.