Interpretation of miscibility curves
This is no different from what is observed in many binary systems. Here, the phenol and water exhibit a miscibility gap. As you noted, above point C, the liquid is one liquid with whatever composition it has. If one were to have a liquid at the composition corresponding to C, at a temperature above 66C. you would have one liquid. As you dropped the temperature to 66C, nothing would change. 

Drop below 66C, and the liquid will phase separate in to two liquids of different compositions, given by the points on the miscibility curve. To figure out how much of each liquid you have, you do the appropriate weighted average as given by the book. That weighted average can be seen to be correct, since, say at 20C, sitting on point A you would be 100% liquid with the A composition, and sitting on point B you would be 100% liquid with the B composition. In between, you are balancing between the two liquids, which are in thermodynamic equilibrium.

If you have a book of phase diagrams handy, look at almost any phase diagram involving Pb, and you will likely find a liquid-liquid miscibility gap. For fun, look at Ag-Ni, two metals with fcc crystal structures, and you will see that the solids are immiscible, and there is even a liquid-liquid miscibility gap.