In Einstein's general relativity the structure of space can change but not its topology. Topology is the property of something that doesn't change when you bend it or stretch it as long as you don't break anything.
If I take the theory as we have it now, literally, I would conclude that extra dimensions really exist. They're part of nature. We don't really know how big they are yet, but we hope to explore that in various ways.
I wouldn't have thought that a wrong theory should lead us to understand better the ordinary quantum field theories or to have new insights about the quantum states of black holes.
Having those extra dimensions and therefore many ways the string can vibrate in many different directions turns out to be the key to being able to describe all the particles that we see.
Even before string theory, especially as physics developed in the 20th century, it turned out that the equations that really work in describing nature with the most generality and the greatest simplicity are very elegant and subtle.
As far as extra dimensions are concerned, very tiny extra dimensions wouldn't be perceived in everyday life, just as atoms aren't: we see many atoms together but we don't see atoms individually.
As of now, string theorists have no explanation of why there are three large dimensions as well as time, and the other dimensions are microscopic. Proposals about that have been all over the map.
As for the forces, electromagnetism and gravity we experience in everyday life. But the weak and strong forces are beyond our ordinary experience. So in physics, lots of the basic building blocks take 20th- or perhaps 21st-century equipment to explore.
It's indeed surprising that replacing the elementary particle with a string leads to such a big change in things. I'm tempted to say that it has to do with the fuzziness it introduces.
Technically you need the extra dimensions. At first people didn't like them too much, but they've got a big benefit, which is that the ability of string theory to describe all the elementary particles and their forces along with gravity depends on using the extra dimensions.
But the beauty of Einstein's equations, for example, is just as real to anyone who's experienced it as the beauty of music. We've learned in the 20th century that the equations that work have inner harmony.
On the other hand, we don't understand the theory too completely, and because of this fuzziness of spacetime, the very concept of spacetime and spacetime dimensions isn't precisely defined.
So when you ask me how string theory might be tested, I can tell you what's likely to happen at accelerators or some parts of the theory that are likely to be tested.
