Nib asked, and Nib — askNib's tutor — drew the answer live at a whiteboard. This is the spoken transcript; enable JavaScript to watch it drawn.
why does time slow down near the speed of light
First, a correction: there's no master clock that slows. Each of you sees the other's clock run slow. To see why, build a clock from a light pulse bouncing between two mirrors, a distance h apart.
Now watch that same clock fly past you. While the pulse is in flight the clock slides forward, so you see a longer diagonal path. But light's speed is the same for everyone.
Half a tick makes a right triangle. Up is the height h. Along is v times t over two. And the slanted light path is c times t over two. Pythagoras does the rest.
Solve it: the moving tick t equals the rest tick t zero over the square root of one minus v squared over c squared. That factor, gamma, is never below one.
It's real. Cosmic-ray muons are born about fifteen kilometres up and live just two point two microseconds. At nearly light speed that's six hundred sixty metres, yet about a quarter reach the ground.
And your phone uses it. A GPS satellite's clock runs about seven microseconds a day slow from its speed alone, so the system corrects for relativity every day.
Time doesn't slow in your own frame; it's the speed of light staying fixed that stretches the other fellow's ticks.