Traveling close to the speed of light (according to the theory of relativity, no physical object can actually travel at the speed of light) does indeed change that principle, insofar as that what takes 25 years from the perspective of everyone else outside the spacecraft, takes only (say) 4 years from the perspective of the people and things inside the spacecraft. I don't fully understand the reasons why myself, I just know for certain that that's what physicists universally accept.
Apparently there've been experiments of various kinds to demonstrate this. One of the better known (if I recall correctly) involved sending planes around the world with atomic clocks on board, and comparing them to atomic clocks on the ground. The clocks on board the planes experienced a slightly shorter (like, microseconds only) passage of time than the ones on the ground.
Right. That's the Hafele and Keating Experiment. More familiarly, time dilation is regularly observed when sub-atomic particles have predictably extended lives after being accelerated.