edited for clarity.
The reason this principle of relativity holds true is based on empirical observation of muons entering Earth's atmosphere, which travel at very near the speed of light, however have a distinct atomic decay resulting in a measurable and invariant half life.
For example, the muon may live for say, half a second as is travelling at 87% the speed of light, which has a time dilation/relativistic mass factor of x2.
At 87% the speed of light an object could travel, say, 275km per second (rough hypothetical here). The muon should be able to travel around 138km in half a second, but the muon actually lives for a full second.
Yet measurement of its atomic decay within its frame of reference still yields a half second atomic decay.
The inferrence is the muon experiences a full second of relativistic life as a half a second within its own reference frame.
Speed is a measurement of distance/time (inherently relativistic). To the muon time does not change, the distance it has travelled within that time has doubled.
Cut to interstellar travel.
Travelling at 87% the speed of light (quite attainable by rather expensive feats of engineering), a Lorentz-factor of 2, halves the passage of time for shipboard occupants relative to Earth.
Example: a star system at 50 light years distance is identified to have extrasolar, potentially habitable planets (by spectrography indicating ozone and nitrogen), so an exploration team is sent to investigate in a craft capable of 87% the speed of light. It would be roughly a 60 year journey which would require at least two full generations of successive occupants to make the journey.
But due to the Lorentz-factor of time dilation, the journey only takes 30 years for the occupants within the spacecraft, making the journey possible within one lifetime.
For all intents and purposes and especially among a space based civilisation, to the occupants, 50 light years has been covered in 30. They have effectively travelled well beyond the speed of light.
It is a fact however, that the universe around them will continue to change at its own rates. Thus it is more corroberative to say they had experienced length contraction in their direction of travel rather than had travelled faster than the speed of light.
Current physics does not prohibit travel at any speed other than lightspeed itself. A Lorentz factor of 3 is achieved at 94%. At 97% it's 4 and at 98% it's 5. That's 150 light years travelled in the space of 30. It keeps going up sharply like this (parabolic), keeping interstellar travel quite possible within the lifetimes of the crew at speeds very close to the speed of light, without breaking this "universal speed limit."
Thus interstellar travel within the human lifespan, according to physics at least is quite possible, just ridiculously expensive. It should be noted that according to some academics, it is so expensive in fact that no civilisation could hope to achieve it.
The jury's still out on that one in the scientific community however.
This effect is known as length contraction, which is the inherent flipside to time dilation.
The "lengths and measuring rods" analogy which Einstein provided in his book on Special Relativity is designed to be purely illustrative of a theoretical concept, rather than a hypothesis of an actual physical examination of objects, for which you would have to be in the same frame reference to begin with...
It was designed to shake up existing Euclidean assumptions about spacetime. All length contraction is perceptual, but perceptual (ie. bodies of reference) is the same thing as reality in the physical world and that is a very impacting assertion, the kind of thing which scares disciplinarians and conservatives to death because of its philosophical connotations.
