He actually explained it quite simply. The faster you go, the slower you move in time. If you reach the speed of light, time doesn't pass for you at all. But only photons can move that fast as they have no rest mass.
For those interested a bit more in this, there is pretty nice example of the passage of time getting slower with increasing speed. There are particles called mions, they're similar to electrons, but heavier. Their mean lifetime is about 2 . 10^(-6) seconds (that means VERY short). Mions can origin in upper layers of the atmosphere due to its interaction with cosmic radiation. Because they exist so briefly, we shouldn't detect these on the surface, but yet we do. This is because mions travel and speeds close to light, so even though only 2 . 10^(-6) secs pass in their own time, we can observe them for several seconds.