I had a bit of trouble solving the full-street AKQ game without reading through the chapter.  They do a lot of things without really explaining them (which is fine) or justifying their method. 

Is the basic idea here that:

1)  Poker is a zero sum game, and so each strategic option has a corresponding strategic option for the opponent (i.e. the amount of hands we bluff corresponds to the amount of bluff-catchers that our opponent calls). 

2)  If each player plays each given situation in such a way that their opponent will not be able to uni-laterally increase their expectation by changing the value of the "corresponding" variable, then we will have a series of mini-equilibrium(s).

3)  If we can then solve each player's value in the game in terms of an "asymmetric" variable (which only one player controls), then all we need to do is find the values of that variable which give max-value strategy and we have solved the game.  This is because, due to the "mini-equilibria", neither player can improve their value by changing one of the symmetric variables, and the player which controls the "asymmetric" variable cannot unilaterally improve their expectation by playing a different value of this variable, since any change they make would result in lower value.  Therefore, we have accounted for all possible strategic options in such a way that we are by definition, at (Nash) equilibrium.