These "dimensions" aren't real directions you can travel in. They're really mathematical constructs to make their equations work better.
For example, we can't move back and forth in time at will, but we still consider time a "dimension" on par with the three physical dimensions, because it's easy to consider them all the same for math. You may have heard that gravity bends "spacetime" - that is, it distorts time the same amount and way it distorts space. So it's easier to consider time just another dimension of space, so you can use the same equations.
They really don't have proof - at that point it's just mathematics. But it does have a use. For example, it was Einstein's thought-experiment about what happens if two spaceships approach each other near the speed of light, that led to his mathematical exploration and theory of relativity. Now that we know relativity, we can use it to make our space flights more accurate. If we didn't think about the math first, we would just assume that our instruments are just off by a fraction of a percent. If Einstein hadn't done his math first, the error would be too small for us to wonder if there's a mathematical explanation. But correcting for the error saves us a lot of worry on space flights.
A lot of physicists are working to the Grand Unified Theory, which is a name for a single theory that covers the actions of all forces and all particles. In the "GUT", each particle and force is just a special case of some super-object that has an easy mathematical explanation and can do everything. This 10-dimension plan (I've even seen up to 26) is a way of allowing superstring theory. Each superstring is a small object that vibrates in 10 or however many dimensions, and depending on how it vibrates, it acts like a different particle. Finding the GUT equation is the Holy Grail of physics, and it allows us to explain why our universe works (as opposed to how, which we pretty much have now).
We're at a crossroads like we were at the end of the 1800s. They figured every problem had been solved, except for one seemingly minor one known as the "ultraviolet catastrophe" - if you look at the math a certain way, it seems to say that objects radiate an infinite amount of energy. It was Max Planck solving this puzzle that brought on the entire field of quantum physics and in effect almost all of 20th-century physics.