That is certainly an interesting question! And yes, your reason that we "catch up" with the light is indeed one of the oddities that can only be resolved if time slows down.
This is how it happens:
The speed of light is always a constant to any observer, and let's call it c. What's so significant about that? It is actually very odd. Imagine a person, say Mary, in a rocket moving at v meters/second. She is holding a flashlight pointed to the front, and she turns it on. So to Mary, light rushes off away from her at a staggering speed c. This makes sense, because to Mary, light has to travel at speed c. Now imagine that John is standing still and looking at Mary's flashlight. According to John, the light does not rush of at a speed of (c + v)meters/second, but just c!
Something must be wrong – a ball thrown forward by a passenger in a car must move faster than one thrown by a stationary guy because the car is moving. But light does not behave like ordinary objects. According to both Mary and John, light travels at speed c, no more and no less.
So how does the fact that time slows down come into the picture? According to Einstein's Theory of Relativity, it turns out that both time intervals and distances depend on the motions of the observer. To John, Mary's spaceship is shorter (in its direction of movement) and Mary's watch ticks slower, i.e. her time passes at a slower rate. Interestingly, to Mary, John's "distances" are also shorter and his watch also ticks slower. This is because of the fact that both Mary and John's points of view are equivalent – both are moving relative to the other, and no point of view is more "special" than the other. After accounting for these effects of what we call time dilation and length contraction, light moves at the same speed for both observers. This video
demonstrates the concept neatly.
Hope this helps!
(published on 12/16/2010)