This is known as time dilation and was discovered by a mathematician named Lorentz.
It is best explained by the light clock visual example:
http://home.earthlink.net/~chkingston/LightClock.htmhttp://en.wikipedia.org/wiki/Light_clockBasically let's say there is a cart with your friend on it. The cart has a light clock on it which consists of two plates and a photon of light bounces up and down between the two plates. Up and down once counts as one second.
You and your friend calibrate your watches to the light clock and then your friend in the cart takes off and travels at something like half the speed of light.
As your friend on the cart goes speeding by, to you, the photon no longer appears to travel up and down, because it's moving horizontally at the same time, since the cart it is on is moving. So from your perspective, the photon takes a diagonal path up and down, and if you traced it you'd see the path forms zigzaging triangles. like /\/\/\/\/\/\/\/\/\/\/\
If you know some basic trigonometry you know that the slanted part, the hypotenuse, of a right triangle is longer than the legs of the triangle. So from your perspective the photon takes a longer path.
Since the speed of light is a constant, the photon can't speed up to strike the plates at the same time your watch ticks. it takes longer to do this because it must travel a longer path from your perspective. So when you compare your watch to the light clock, the light clock is running slow.
But wait, to your friend on the cart, the photon still travels up and down because he is traveling with the light clock, and his watch ticks with the light clock. This means, from your perspective your friend is running slow too. In fact, everyone on the cart looks like it's running slow!
Not only that, but to your friend, his watch is running fine and everyone on the ground that he is passing on the cart looks like it's running slow, because if the light clock were on the ground, he'd see the photon take the longer path.
Now why does time stop at the speed of light? It doesn't really, the actual equation describing all of this is "undefined" at the speed of light. It can't give any useful information for what happens at that point. But it can tell us what happens infinitly close to that point, an what happens is the closer you get to the speed of light, the slower time goes.
Back to the light clock. The faster you see the cart go, the the shallower the angle the path of the photon takes becomes, and with this, the longer the path becomes and the longer it takes for the photon to strike each plate from your perspective. As the cart nears the speed of light, things start to move so slow on the cart from your perspective, it looks as if time has stopped on the cart. Your friend is nearly "paused" as is the light clock.
From your friend's point of view, you and everyone on the ground looks much the same way.
I hope this helps. The wikipedia link will show you the math if you should care to see it.