Alice and the assembly of birds and the mouse are all wet upon the shore. A good deal of confusion erupts over how they should get dry. Finally the Mouse decides the best way to make everyone dry is to tell a very dry story (that is, he tells an exceptionally boring story about English history). This ironic play on words, however, does little to dry anyone so a new plan is devised by the Dodo to hold what he calls a Caucus Race, where there is little or no course and everyone can start whenever they want. The whole group participates and they run and run and run until they are dry, at which point the Dodo declares the race over and everyone a winner. Then he demands that Alice award the prizes. She finds a box of snacks in her pocket and hands one piece out to each winner, though she has no prize left for herself. The Dodo asks her if she has anything else, and she produces a thimble. In a brief but solemn ceremony the birds bestow her thimble back upon her as a prize.
Now that they are dry, the Mouse tells his tale, which Alice imagines is shaped like a tail. Essentially, the tale is a short poem printed on the page in the shape of a mouse tail. Once the tale is told, Alice and the Mouse have a short fight over how the tale should have worked out. Then, once the fight has ended badly, Alice wishes aloud for Dinah to be there. All of the birds then politely ask Alice who Dinah is and, again, Alice goes on about how good Dinah is at catching and killing mice and birds. Offended and frightened, the party of animals politely excuse themselves and Alice is alone again.
Yet again, Carroll is making a case for adulthood. In the first half of the story there is this Caucus Race, essentially a game with no rules. What Alice recognizes and what Carroll hopes the reader sees is that without rules, no amount of play is really fun. It may get you dry and it may make you tired, but in the end it probably isn't much fun. Games are competitions and competitions require rules and discipline. Without the discipline of adulthood, life will start to become rather pointless, filled with empty ceremonies and meaningless accomplishments.
The second half of the chapter is in many ways a reiteration of the previous chapter. Alice perhaps recognizes that games need rules so that they make sense, but she hasn't yet learned that rules are needed for all occasions. Conversation, to be meaningful, needs rules just as a game needs rules. And in order to successfully hold a conversation a person needs boundaries. Alice has not learned that talking about her cat Dinah to Birds and Mice who are potential meals of Dinah is Out Of Bounds.
In short the first half of this chapter is used to prove that Rules Are Necessary. The second half of this chapter drives home the point that If Rules Are Necessary, then it is important, vital perhaps, To Learn The Rules And Follow Them.