Nothing is definite in science, e.g. Newton's laws survived for a couple of centuries before we started finding things that didn't fit and discovered that they weren't the whole story. Extra dimensions currently come into the "no evidence as yet" category.
That analogy was very nicely crafted (not by me!) to appeal the the (tory) politician who was offering a bottle of champagne as a prize
. I suppose could offer a different analogy (filling the universe with treacle, perhaps?). But the basic idea is that there is some quantum field whose average value in "empty" space is non-zero, and which interacts with other particles. Those other particles gain energy as a result of this interaction, and hence mass (E = mc^2 means that if a particle at rest has an energy E, that is equivalent to a mass m = E/c^2). The Higgs mechanism, or something like it, is necessary because just adding mass terms to the field theory destroys an important mathematical property which makes the theories actually usable. Doing it this way, via interactions with this extra field, preserves that property. Whether that is what actually happens in nature we don't yet know - if not, there must be some other process that produces the same effect, and that should also be accessible to the LHC. Arguably it will be more interesting if it turns out not to be the Higgs (it would spur more new thinking and some re-evaluation), but it's too early to say yet.
As for interstellar travel, it depends on how close to the speed of light you travel how much times has elapsed. But it's actually the other way round: if you travelled at 0.9*the speed of light, and we overlook the time taken to accelerate/decelerate, then 10 years would have elapsed on Earth but only 4.4 years for you. If you travelled at 0.95*c it would be 9.4 years on Earth but 3 years for you, and at 0.99c it becomes 9.1 years vs 1.3 years to you. To the observer on Earth it appears that you travel 4.5 light years each way but your clock is running slow. To you it appears your clock is normal but the distance is reduced (Lorentz contraction).
And as evidence that this is a real effect, we're both being hit by a couple of cosmic rays each second (easy to see them with the right apparatus). These particles are muons, produced in the upper atmosphere. But muons have a lifetime of 2.2 microseconds, so should only be able to travel about 700m before decaying (at speed close to c, but without time dilation). They only reach the ground because they are travelling fast enough that time (as we see it) is running slow for them.