Why sum of two binary numbers cannot be determined in $NC^0$ but in $AC^0$?
In an $\mathsf{NC^0}$ circuit, every output bit depends on a bounded number of input bits. But the $k$th bit of the output (counting from the LSB) depends on the first $k$th bits of each input.

To see that $\mathsf{AC^0}$ circuits can compute addition, we need to produce such a circuit. Hopefully you have seen such circuits, and otherwise perhaps you can construct them on your own. Give it a shot.