Although it may not help much, and although it's sneakily substituting constructive for classical logic, I enjoy thinking about this as a computer-science theorist might: namely, interpret the variables $P$ and $Q$ not as statements but as sets, and $P \implies Q$ as the set of functions from $P$ to $Q$. Then we have richer information than mere truth (the set is non-empty) and falsity (the set is empty), but can return to the mere truth perspective when appropriate.
Now $\lnot P$ implies that $P$ is uninhabited, i.e., as a set, there's nothing in it. (Here's where I'm being a little sneaky; you asked about $P$ being false, and I'm talking about its negation $\lnot P$ being true. These are equivalent from a classical point of view, but a constructive point of view requires the finer distinction that I am making.) Then, you might think, there is no function from $P$ to $Q$, so that $P \implies Q$ should be false; but actually that's not quite right. A function, remember, just has to tell you what to do with the elements of its domain, and if the domain doesn't have any elements then there's nothing to do; so there is, trivially, a (in fact exactly one) function from the empty set to any other set. (This is true even if the other set is empty; we never even look in it.) That is, $P \implies Q$ has got some elements (one of them), and so, from the Boolean point of view, is true.
Contrast this with the situation where $P$ and $\lnot Q$ hold, i.e., there's something in the set $P$ but nothing in the set $Q$. In that case, there's no way to specify a rule for assigning elements of $Q$ to elements of $P$, because there are no elements of $Q$ to assign; so $P \implies Q$ is false in this case, as your textbook says.