Ok, here is the straight goods of an easy to understand explanation of how time dilation due to velocity works. Ok, now picture if you would (as in this picture) a spaceship that has two mirrors on either side of the cargo bay, and for arguments sake the mirrors are one meter apart, so the photon of light has to move two meters to get there and back. Now if the shuttle is traveling at half the speed of light, the shuttle will move one meter sideways as the photon is moving between mirrors, now the light is no longer traveling in a straight line to an observer off of the ship, but in a triangular shaped line, therefore the photon is now traveling a longer distance, and will take a longer time outside of the ship to complete its journey. Check out this page.
http://musr.physics.ubc.ca/~jess/p200/str/str9.html
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