Moving clocks tick slower in the lab frame
Time dilation follows directly from the constancy of light speed. If a clock is moving at velocity v, the time interval it measures between two ticks is shorter than the time interval measured by a stationary observer watching that clock. The Lorentz factor gamma = 1/sqrt(1 - v^2/c^2) quantifies the slowdown: a moving clock runs slower by a factor of gamma. At 0.6c, gamma is 1.25, so a moving clock ticks 20% slower. At 0.99c, gamma is about 7, meaning that moving clock runs roughly 7 times slower compared to a stationary observer's clock. The symmetry of relativity means a stationary observer's clock also appears slowed when viewed from the moving frame, so there is no contradiction: each observer sees the other's clock as slow.
GPS satellites and muon detectors verify the effect
GPS satellites orbiting Earth at high altitude and high speed experience time dilation relative to ground clocks. The satellite clocks run slightly faster (they experience weaker gravity and move relative to Earth's center), so the GPS system must account for relativistic corrections totaling microseconds per day. Without this relativistic correction, GPS positions would drift kilometers within hours. Atmospheric muons, created by cosmic rays and moving at nearly light speed, live longer in the lab frame due to time dilation (their decay lifetime is dilated by gamma). They can traverse the entire atmosphere before decaying, whereas static muons would decay within a few microseconds. This 'muon paradox' disappears once time dilation is included.