One interpretation of $KL(P,Q)$ when P and Q are defined over a discrete space comes from encoding.
When you generate data according to P and you want to compress that data, like you would in a .zip file, you would like to use $\log(P(i))$ ressources to represent symbol $i$. That would be the most efficient way of representing data generated according to $P$.
What happens when you code instead symbols as if they were generated by $Q$ instead ? You then use $\log(Q(i))$ ressources to represent symbol $i$, which is less efficient. The difference in efficiency is $KL(P,Q)$
So, coming back to your question, it makes sense that symbols that never appear $P(i) = 0$ in the sequence we're looking at, do not affect the value of $KL(P,Q)$. Since they never appear, the cost of encoding them is 0