"If $P$ then $Q$" tells you that whenever you know that $P$ is true, you may automatically conclude that $Q$ is true.
"If not $Q$, then not $P$" tells you that whenever you know that $Q$ is not present, you may conclude that $P$ is not present.
If ($P$ implies $Q$) is true, then knowledge of the presence of $P$ always gives us that $Q$ is present. If $Q$ isn't there, then there must be no $P$ to have put it there. Likewise, if the absence of $Q$ automatically gives us the absence of $P$, and if $P$ is present, there can be no not-$Q$ to have given us a not-$P$.
Similarly, either we have $P$ or we don't have $P$. In the case that we have $P$, we are guaranteed to have $Q$. The other case is that we don't have $P$. So we either don't have $P$, or we have $Q$ (which we got by having $P$): not-$P$ or $Q$.
On the other hand, let's say all we know is that either $P$ isn't true or $Q$ is true. If $P$ isn't true, then $P\Rightarrow Q$ is trivially true. Let's deal with the case where $P$ is true. So we know that $P$ is true and also that either $P$ isn't true or $Q$ is true. That is, since $P$ and not-$P$ can't both be true, if $P$ is true, then $Q$ is true.