Taking the speculation in the article at face value, and thus assuming that NASA has found an arsenic-based lifeform in a shadow biosphere on Earth, here's why it's important:
All life on Earth that we know of is related. It all uses the same basic DNA/RNA mechanisms (including the same four base pairs), uses the same specific molecules that prominently feature carbon as the basic assembly blocks of the cell, etc. To use the ever-popular car analogy, cars can look quite different from each other, but they're all still essentially made out of the same things: bolts, gears, copper wiring, etc.
Well this other kind of life is completely different. It's so different that we know it cannot possibly be related to all of the other Earth life that we've known about thus far, as there is nothing in common. That means abiogenesis (the spontaneous generation of life from precursor non-living materials) happened at least TWICE on just this one planet.
So while this isn't extra-terrestrial life, it does have all sorts of potential ramifications on the potential existence of extra-terrestrial life. Before today, it was possible to speculate that one solution to Drake's Equation was simply that spontaneous generation of life was so rare that it only happened once, ever. But if we now found that it's happened multiple times just on this one planet ... then hell, it could be happening everywhere, all the time.