Ever since I read The Elegant Universe years ago, I've had a number of questions related to this (as I imagine many people have). This is the first time I've seen the topic discussed by professional scientists, though, as opposed to people like myself with a hobby interest in the subject or in science fiction (Alastair Reynolds makes use of it in one of the Revelation Space novels, for example).
For the most part, it seems like String/M-Theory is very difficult (at best) to test using technology we have access to at present. But because it includes the idea of gravity being a force which can travel between branes, it's seemed to me and a few friends of mine that this would definitely produce some interesting effects in the real world.
As the article discusses, there should be some subtle evidence of the effects of gravity from external sources on the large-scale structures of our own universe. I would think maybe even enough to at least partly explain "dark matter" and "dark energy", since those are basically the known matter in our universe behaving as if there were a lot more mass that we can't actually see (one set to hold relatively closely-spaced matter together, and the other to accelerate the expansion of the large-scale structures away from each other, if I understand correctly).
A simple flatland-style analogy for "dark energy" might be that our universe is a sheet of paper which is intersected by a universe which is wrapped around into a tube shape or a torus. The gravity of the mass in that second universe pulls objects in our universe toward it, so for the part of our universe in the "eye" of the tube, they tend to accelerate away from each other. That's a vast oversimplification, but I'm not a physicist
For "dark matter", the idea that's always stuck with me since reading The Elegant Universe is that maybe some/all of the most massive objects in our own universe - especially the black holes at the centers of galaxies - are caused by the same kind of cross-brane effect. If you have a bunch of matter clumping together in one brane/universe, and it exerts gravity which can cross into other branes, then it seems like it would create corresponding accretions of mass in other nearby branes. Basically, that what we perceive to be a roughly spherical/point object would effectively be the hyperdimensional equivalent of that same shape that would "pin" itself together across branes.
Where I see this as becoming testable (and I could be wrong - again, I'm not a physicist) is that if this were the case, there should be examples of anomalous astrophysical objects and events, where the mass we observe does not line up with effects we also observe. For example, a stable neutron star suddenly flashing into a black hole when it passes too close (hyperdimensionally, of course) to a large mass in another brane. Another example might be a star or planet whose mass can't be reconciled with its observed size - e.g. maybe there is a planet the size of our moon, but which exerts gravity as if it were made entirely out of a material ten times as dense as uranium.
I know that in the context of our own universe/brane, there's no way to pull matter out of a black hole (other than Hawking radiation), but assuming the "hyperdimensional singularity"-type thing I described above is accurate, would it be possible for the cross-brane components to separate (since they wouldn't actually be touching, just exerting gravity on each other)? If so, there might be even stranger observable effects, like neutron stars that "flash" into black holes, but then return to their former state when the mass in the other brane(s) is pulled too far away. IE they would "blink".