I don't see how it changes. If I slow down my car it will take me much longer to reach the same destination. Since speed = distance over time, I could say the time or the distance was "stretched". It can be "true" depending how you look at it. I don't see how in reality the distance changed.
I think the point in Relativity is it's different depending on from where you look at it.
It's been quite a while, but my memory is that the significant change is that for Newton the "clock" is the same everywhere for all observers - a uniform, absolute, independent attribute - the same everywhere in the universe when viewed from anywhere else in the universe. Space and time could be considered separately.
For Einstein, the speed of light is that way, and spacetime a continuum.