 Solution

For each nonnegative integer $z$, the events $\{X=z-y,Y=y\}$, $0\le y\le z$, are pairwise disjoint and their union is $\{X+Y=z\}$. By countable additivity and then independence,This is the discrete convolution of the two probability mass functions. If the convention for natural numbers excludes zero, set both probability mass functions equal to zero there; the formula remains valid. Also $p_Z(z)=0$ for negative $z$.