The problem opens with
Suppose P and Q are each true.
You then think about the possibility that
P can be false.
But P can't be false - the assumption is that P is true. The mistake you're making is thinking in terms of possibility, e.g. that P could be true even though it isn't, and then interpreting "If not P, then not Q" as a statement about possibilities. But that's not the sense being treated here. P is true, Q is true; so "not P" and "not Q" are both false. The question, then, is: how do we think about "false implies false"? (Or even worse, "false implies true"?)
Ultimately this comes down to how we define implication in propositional logic, but the point is that "false implies ---" is vacuously true; it might be easier to first think about why a statement like "Every purple flying elephant is the king of France" could be considered true.
One important takeaway here is that we're not thinking of "implies" in terms of causality or possibility. If you want to talk about such things, we have to go beyond propositional logic - modal logic is a good place to set up shop.