If no frame of reference has precedence over any other, why do we say that if a spacecraft leaves earth and attains a significant percentage of c, time would pass more slowly for the crew of the ship than for those still on earth?

From the ship's perspective, it's the earth that's speeding away at tremendous velocity, no? So why then doesn't time pass more slowly for the earth, as the crew of the ship age at a relatively accelerated rate?

So if the ship returns to earth, which has "aged more" relative to the other? Why the difference between the two frames of reference? Can the reason be explained in layman's terms? (simple equations only, please) :-)

Thanks to any and all takers...