Page 136 says Following Step 1 of Definition 7.5, the top of Figure 7.23 shows $A_4$ organized by the subgroup $\langle x, z\rangle$ (which is isomorphic to the Klein $4$ group. This reorganization required using arrows for the generators $x$ and $z$, as well as a generator that connects cosets of $\langle x, z\rangle$, in this case $a$. Step 2 of Definition 7.5 asks us to collapse each of the three left cosets of $\langle x, z\rangle$ into a single node, uniting any arrows that are thereby rendered redundant.
This is from Nathan Carter page 136 figure 7.23 Visual Group Theory.
Question 1. I don't understand how these Cayley diagrams represent $A_4$. What do the blue, green, red mean?
Question 2. Where's subgroup $H = \langle x, z\rangle$ that's isomorphic to the Klein $4$ group? What color(s)?
EDIT @Richard Peterson 2/2/2014 ----
Question 3. How do you see $A_4 = ($ any of the 3 cycles under $ )V_4$ without actually calculating all the permutations in orange underneath?
I know $A_4$ is
Take x and z to be any two of the three nonidentity permutations in V4 of the subgroup lattice you provided at the bottom of your message. V4, (as you know) is the Klein 4 group. It is the top square in each of Nathan Carter's diagrams, with e being the identity and y equal to xz, y being whichever nonidentity permutation in V4 that you DIDN'T choose when you chose x and z.
Take "a" to be ANY three-cycle element in the group A4. As user86418 said, the nodes are group elements and the colored arrows represent (left) multiplication by the group elements x (in green), z (in red), and $"a"$ (in blue).
So, for example, the blue arrow from node y pointing to node d means ay=d. Each of the three squares in the diagram are cosets of V4. V4 is called H. The top square is V4=H=eH as stated above, the right lower square is aV4=aH, and the left lower square is aaH....I may have overexplained parts of this, but some other person besides you may find the extra explanation helpful... (Note the reason I put "a" in quotes was to try to be unconfusing, hopefully I wasn't more confusing--I did it to mark $"a"$ as ANY group element in $A_4$ when it wasn't the one letter English word a, except when I multiplied by the group element "a", like aH.)