As far as Mr. Flat Hat is concerned, the light beam took longer to make the round-trip, as it has had to travel further. Our everyday experience of time is that it goes by at a constant rate, but this isn't what actually happens! If we accept Einstein's word that all observers obtain the same value when they measure the speed of light, some other 'constant' has to give, and in this case it is time.
This is true for all clocks, not just those based on a bouncing pulse of light. Even a digital clock or a grandfather clock would appear to run more slowly - it is a simple fact that moving clocks run slower when observed by someone in a different inertial frame.
This phenomenon has actually been observed using atomic clocks accurate to 0.0001 nanoseconds flown round the world in supersonic aircraft!
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