There's another way of looking at the time-travel special relativity problem by the way. It's a bit more of conceptual leap, but I think it's worth while.
Rather than just travelling through space (as in, when I walk to the fridge I travel through space), we actually travel through something called spacetime. Ok, so this is a bit weird to understand so let me see if I can explain it. The universe is made of three spatial dimensions and one temporal dimension. That is the fabric of the universe itself (according to Einstein. Though this has been quite broadly refuted but still explains it well enough).
We are constantly in motion. So if we're not moving through space, our motion is devoted to time. Rather conveniently, the "speed" of our motion is equal to c, the speed of light. When we start moving through space, we move less quickly through time. So when Usain Bolt sprints at 40kph, he takes 40kph of his time motion and devotes that to space motion. So time moves c-40 more slowly.
Returning to our space ship, when it goes cruising away at 99% of the speed of light, 99% of its motion becomes devoted to space and only 1% to time. Whereas on earth, all the people are basically stationary (moving at mach 1 still only changes time by 1 billionth of a second if you circumnavigate the world, so we can make the approximation that everyone is stationary). So the practically all our motion is devoted to moving through the time dimension. So there's a huge discrepancy. The spaceship is travelling through time much more slowly than we are, and thus, once it returns there will be a huge difference between the time observed to have pass by the astronauts and the time observed to have passed on Earth, just as we described above