The reason that falsifiability is a core requirement of science is because if there is no way a proposition can possibly be falsified, then there is no way to objectively assess whether or not it is true.

This is not to say that the proposition is false. It's possible that things can be unfalsifiable and true nonetheless, but those things would still exist outside the range of the scientific method (at least until/unless our understanding of reality expands enough to devise a test). That's an intentional trade-off, in order to gain greater confidence in the truth of the things we can test.