I am not an expert and it's been a while, so please feel free to correct me.
There are two contexts the string theory is talked about. First, it is by most considered the most promising theory of fundamental interactions (a.k.a. the theory of everything in press and blogs.) Second, it has been shown about ten years ago that there is an equivalence between some version of string theory and some field theories. The second part makes it possible to use string theory for calculations in quantum field theory. It is not related at all to whether string theory is indeed the correct fundamental theory. (It also means you can't kill string theory: even if some other theory, say loop quantum gravity, is the correct fundamental theory, you can still use string theory for calculations in the quantum field theory.)
As far as I understand, the new results the OP mentions are derived using the latter technique, i.e., applying string theory to obtain results in regular field theory. As such, it has nothing to do with demonstrating physical reality of string theory, at least not in the most obvious sense.
My limited understanding of String theory suggests it is closer to a scripting language for describing possible universes than a unified theory. It's interesting as a branch of mathematics, but you can make it say just about anything by playing around with some assumptions.
IMO, saying string theory applies to some process is closer to saying you used Pearl to model fluid mechanics instead of discovering a fundamental law of the universe.
PS: Which is not to say it's useless. Having a better model for high temperature super conductors opens many doors. With proper verification this could easily be Nobel Prize worthy.
Einstein swore that his very own equations dealing with quantum mechanics had to be wrong because it described a phenomenon known as quantum entanglement, which goes against all logic and seems impossible. This led him to believe that the equations were almost right, but with an error somewhere. To quote him, ""I find the idea quite intolerable that an electron exposed to radiation should choose of its own free will, not only its moment to jump off, but also its direction. In that case, I would rather be a cobbler, or even an employee in a gaming house, than a physicist"
It wasn't until a few decades later that we observed quantum entanglement in a lab... showing that the equations had in fact been right, even though Einstein had thought they were wrong.
The story in the op is a little different. A weird phenomenon is observed but unexplained... now an equation explains it, but you have to wonder if the equations were made to fit the observation. The best way to test this is to use the equations to predict some unknown and unobserved outcome, then see if the actual result meets the expected result.
If they mean the words how I understand them, you can explain something without making accurate predictions.
If I walk by a box and it falls over, I can theorize the the wind did it. That "explains" what happened. But it doesn't make my theory true.But say I predict that next time I pass it the wind will knock it over. So I walk by it again — it doesn't fall — then my explanation is incomplete.
That's why we require predictions. The real test is how accurate they are.
To expand on what the other posters have said, humans are natural storytellers. We want things to have a narrative, and we are bad at determining if a theory merely fills our desire for a narrative. Starting from a known phenomenon and crafting a theory around what we know should happen has the danger of creating a model that is too specialized or completely contrived.
If, however, the theory is able to make predictions about things we didn't already know, and those predictions agree with experimental data, then there's a higher chance our new model maps to physical reality.
I wouldn't quite say that because it implies we must have the ability to test it. A better thing to say is that a theory must provide falsifiable predictions, even if we don't have the ability to actually test it today.
Ahh yes, the scientific method: If your theory explains the experiment better than other theories it gets more credibility, if your theory doesn't fit with experiment it's wrong.
If only mainstream media would employ the scientific method in their journalism.
Just throwing this out there: is it the first time a calculation based on string theory has been published in Science because the string theory community prefers arXiv?
There are two contexts the string theory is talked about. First, it is by most considered the most promising theory of fundamental interactions (a.k.a. the theory of everything in press and blogs.) Second, it has been shown about ten years ago that there is an equivalence between some version of string theory and some field theories. The second part makes it possible to use string theory for calculations in quantum field theory. It is not related at all to whether string theory is indeed the correct fundamental theory. (It also means you can't kill string theory: even if some other theory, say loop quantum gravity, is the correct fundamental theory, you can still use string theory for calculations in the quantum field theory.)
As far as I understand, the new results the OP mentions are derived using the latter technique, i.e., applying string theory to obtain results in regular field theory. As such, it has nothing to do with demonstrating physical reality of string theory, at least not in the most obvious sense.
The original article: http://www.sciencemag.org/cgi/content/abstract/1174962