These super-heavy elements are fantastically unstable, and decay quickly. What they do is smash nucleii of more stable elements (which he have in abundance) together really, really fast, using particle accelerators. They have previously calculated that smashing atoms of X and Y together at a certain energy will theoretically yield a nucleus of undiscovered element Z. Further, based on the laws of quantum mechanics, we can predict whether Z will be stable or if it will undergo radioactive decay.

All the elements heavier than lead (like uranium) are unstable to a degree and eventually decay, but the really heavy ones are *really* unstable and decay so quickly (fractions of a second) that you’ll never see them in nature. So the only way to see them is to try to make them. And since you can’t really examine individual atoms under a microscope, the way you determine that you’ve seen them is by looking for the expected decay products. If you can set up detectors for those, and you detect them, then that proves that something must have decayed into them, proving that you created at least one atom of the new element. But your detectors may miss some decay events, so you can’t be sure precisely how many you made.

The island of stability refers to a hypothesized range of elements, much heavier than any we have yet created, that is predicted to be stable, with half-lives a lot longer than these useless elements that decay right away. Maybe some of them will even last for years or more. We won’t know for sure until (and if) we can actually make them.