Of course it seems obvious that our world has three space dimensions, as obvious as that the sun orbits the earth. Physics theories typically predict aspects of the world that we do not see. For example, Maxwell’s theory of electromagnetism correctly predicted that the spectrum of light we see was just a part of the full spectrum, which extended into infrared and ultraviolet waves invisible to us.
String theory predicts our world has more than three space dimensions. Contrary to much that is written and said, as I will explain here, string theory is broadly predictive and testable. Before I explain its testability, I will describe why great progress in making a comprehensive underlying theory of our physical world may emerge from formulating theories in more than 3 space dimensions. I’ll call it a "final theory" following Steven Weinberg.
What could we gain by giving up the idea that our world has three space dimensions (3D)? String theory emerged when John Schwarz and Michael Green noticed in summer 1984 that it was possible to write a mathematically consistent quantum theory of gravity only in 10D (10 space-time dimensions). That’s a big gain and clue. For me and some theorists it’s even more important that string theories address all or nearly all of the issues and questions that need answering in order to have a final theory. There has been major progress here in the past decade. The initial excess optimism of string theorists caused an overcompensation, now tempered by increasingly many results. The highly successful and well tested 4D so-called "Standard Models" of particle physics and of cosmology provide powerful accurate and complete (with the discovery of the Higgs boson) descriptions of the world we see, but do not provide explanations and understanding for a number of issues that are addressed by string theory. The success of the Standard Model(s) is strong evidence that sticking with the 4D world gets in the way of going beyond description to explaining and understanding.
To explain our universe, obviously the higher dimension string theories have to be projected onto a 4D universe, a process with the understandable but unfortunate name "compactification" (for historical reasons). Experiments and observations have to be done in our 4D universe, so only compactified theories can be directly tested. Compactified string theories address why the universe is mainly made of matter and not antimatter, what the dark matter is, why quarks and leptons come in three similar families, what the individual quark and lepton masses are, the existence of the Higgs mechanism and how it gives mass to quarks and leptons and force-carrying bosons, cosmological history from the end of inflation to the origins of nuclei (after which the Standard Model takes over), the cause of inflation, and much more. Compactified string theories successfully predicted (before the measurements) the mass and properties of the Higgs boson discovered at CERN in 2012, and make predictions for the existence of "supersymmetric partner particles" some of which should be produced and detected at the upgraded CERN collider in 2015 if it functions as planned. Examples already exist in compactified string theories for all of these. All of this is research in progress, so much still needs to be worked out and understood better, and tested at colliders and in dark matter and other experiments, but we can already see that all these exciting opportunities exist.
In 1995 Edward Witten argued that there was an 11D theory he called M-theory which could give a consistent quantum theory of gravity, and that it could be projected onto several 10D string theories in different ways. They had names like Heterotic or Type II. Those 10 D theories could then be compactified to 4D theories (with 6 small curled up dimensions), and make testable predictions as described above. M-theory can also be compactified directly onto a 7D curled up (G2) manifold plus four large space-time dimensions. The study of such theories is ongoing. The compactified theories are testable in the traditional way of testing physics theories for four centuries. In fact, they are testable in the same sense as Newton’s second law, F=ma. F=ma is not testable in general, but only for one force at a time – for a given force and mass object one calculates the predicted acceleration and measures it. Similarly, the form the small extra dimensions take for compactified M/string theories leads to calculable and testable predictions.
A nice example of how the string theories may help comes from the Higgs boson mass. In the Standard Model the Higgs boson mass cannot be predicted at all. The extension of the Standard Model to the theory called the supersymmetric Standard Model predicts an upper limit on the Higgs boson mass, but cannot make an accurate prediction of the mass. Compactified M-theory allows a prediction (made by me with students and colleagues) with an accuracy of a few per cent, in 2011 before the CERN measurements, and confirmed by subsequent data.
If we want to understand and explain our world, going beyond even a full mathematical description, we should take seriously and work on 10D string theories or 11D M-theory, compactifying them to our apparent 4D world. People often say that string theories are complicated. Actually, compactified M/string theories seem to be the simplest theories that could encompass and integrate all the phenomena of the physical world into one coherent mathematical theory.