Consider the second digit of the binary number. First the inputs A and B are added. Furthermore the transfered information from the first digit, the Carry Bit, has to be added. Hence the circuit gets a third input, the Carry-IN CI. Such a logical unit is called a full adder.
There are many ways to build a full adder using logic gates. The circuit chosen here consists of two half adders with an additional OR gate.
Now back to the problem of two 2-bit numbers to be added.