Why does a moving clock tick slower?

A moving clock ticks slower because light's speed is the same for everyone. That single fact, from Einstein's 1905 relativity, is enough to stretch time.

Here is the thought experiment, the light clock. A pulse of light bounces between two mirrors a distance L apart, and the clock ticks each time the pulse returns to the first mirror. Seen from the clock's own frame, the pulse simply goes up and down. Now watch the same clock fly past you. In your frame the pulse traces a longer, angled path, because the mirrors have moved sideways while the light is in flight. Since the speed of light must be the same for every inertial observer, the longer path can only mean more time between ticks. So the moving clock ticks more slowly.

The clock doesn't have to be made of light. The idea that time dilation applies only to light clocks, and not to mechanical, atomic or biological clocks, is a common misconception. It is time itself, as Einstein showed in 1905, not the machinery.

At everyday speeds the slowdown is far too small to notice: it took optical atomic clocks, in 2010, to catch it below 10 metres per second. Near the speed of light it is significant. For a train at 0.86 of light speed, observers on the ground may measure a 2-second difference on their clocks while the engineer says only one second passed.

One twist: the effect is reciprocal. Both observers measure the other's clock as slower, and Einstein was the first to point out this symmetry. It is a relationship between clock readings, not something you could see by looking at a clock, because light takes time to travel from the clock to your eye.