Time dilation

Proper time is the interval measured by an observer for whom both events occur at the same location. It is the SHORTEST possible measure of that interval; every other observer measures a longer one.

So "time slows down" is a shorthand that misleads. Nobody sees their own clock run slow. What happens is that all clocks moving relative to an observer are observed by that observer to run more slowly than a clock at rest with respect to them — in other words, two observers in relative motion do NOT measure the same elapsed time between the same two events.

This is not bookkeeping on paper. The muon's half-life as measured by an earthbound observer varies with velocity exactly as the equation predicts; and in 1971 Joseph Hafele and Richard Keating verified the effect at low relative velocities by flying extremely accurate atomic clocks around the world on commercial aircraft, with results within experimental uncertainty of the predictions of relativity.

The effect is the twin of length contraction: the same Lorentz factor governs both, and it only becomes appreciable as the relative velocity approaches the speed of light.