The explanation is rather simple if you think of light as particles, like bullets, called photons. One beauty of light is that it moves at a constant rate when traveling through a vacuum or medium. It doesn't move faster or slower unless you change the medium it's traveling in.
Imagine that you're standing at a fixed distance from a clock. As light reflects off the clock, it strikes your eyes at a fixed rate such that the clock's second hand sweeps at the same rate. Time is ticking away like normal.
Now, imagine if you are moving away from the clock at near the speed of light. Since the photos (bullets) are moving at a fixed rate, it will take longer and longer for each particle, reflected off the clock, to catch up with you. Therefore, you'll experience the sweeping of the clock's second hand at a slower rate. Hence, time appears to slow down - the clock looks as if it's running slower.
But, time isn't really slowing down. The watch on your wrist will mark time normally from your point of view. From the point of view of the clock, which is also marking time normally, it will look as if you're slowing down, too.
While this is a naive application of time dilation, since acceleration and deceleration must be taken into account, it really does appear that time slows down when traveling away from an observer at the speed of light. A more interesting observation is the twins paradox where one traveling twin ages slower than his other, non-traveling twin.