Instead of trying to solve the circuit, try to make the circuit from the equations.
In electronics especially, if you directly see the circuit, then you would feel afraid. So this above adder circuit is made from the equation
You can see in page 12-12 that the author has derived some equations regarding the full addition of binary numbers with can consider carry. So in full adder circuit, there are two parts, one for sum and other for carry.
In that it is shown that
So based on above equation, you can understand the above equation.