This Popperian position is contentious (and you have mis-stated it: Popper doesn't think that failure to falsify via experimentation increases the likelihood of truth). AJ Ayer had sound criticisms of it (eg why would we care about experimentation and refutation if we weren't concerned with the truth of our theories?).In the philosophy of science something that is falsifiable technically is something you can only attempt to prove to be false. It's not the same as something that is true (because something that is true, is not falsifiable).
A subtle but important difference to proving something true.
All scientific theory must be falsifiable in order to be science. It must be able to be proven false via empiricism and experimentation. It's never actually possible to prove a scientific theory true (because then it would lack falsifiability), only to experiment on it, fail to disprove it, and make it more likely that it is true (because no-one has been able to disprove it yet).
It's the interesting side effect to the scientific method. In order to uncover supposed objective truths about our existence and the universe, it never ever makes any discoveries about actual objective truth.
Only theories, which much remain falsifiable in order to remain science.
David Stove (the late Australian philosopher) has some interesting work critiquing Popper.