Alright... hold on to your hats. It's going to get technical.
These seemingly unintuitive results about time "changing speed" stem from Einstein's theory of relativity. Let's begin with a though experiment.
Imagine you're beneath deck on an old sail boat. Without looking outside, it is impossible to determine the direction or speed in which the boat is travelling. In fact, this is true for any motion at constant speed. Essentially, if you're blindfolded you cannot determine whether you're moving or whether you're stationary. This phenomenon is called the principle of relativity and has been accepted since Galileo.
Firstly, though, a short history lesson. One of the important questions in physics in the 1800s was explaining the method by which light travels through a vacuum. It was generally believed that light "waves" behaved in a similar manner to sound waves moving through air - ie, light needed some medium through which it propagated. This medium was called the luminiferous aether. The aether was thought to be a completely motionless "thing" which filled all space and could pass through any material object but was able to "carry" light. There were many experiments performed to try and detect the aether, or to try and see its effects. None were successful.
The fact that the aether was completely motionless worried Einstein. He postulated that, should you be able to see the aether, you'd be able to violate the principle of relativity (ie, determine whether you're moving or not). So, locked inside the hull of the boat from our example, by measuring your speed relative to the aether, you'd be able to tell if you were moving. This was one of the arguments that caused Einstein to reject the notion of the aether.
Einstein tried to form a simple picture of how light moves. He imagined a simple set up, where a person was holding a mirror and travelling at the speed of light. Would there be a reflection in the mirror? Maxwell had proven that the speed of light is a constant. If I were holding a mirror, moving at the speed of light, the light from my face wouldn't be able to catch up to the mirror! My image would disappear.
If this were to happen, you could tell you were moving at the speed of light just by looking in a mirror! This would violate the principle of relativity! Einstein postulated (in his paper, On the electrodynamics of moving bodies")
1. The principle of relativity (as set out by Galileo) holds for light as well as ordinary motion.
2. The speed of propagation of light must always be constant, regardless of the "state of motion" of the observer or the emitting body.
In order to prove his theory, Einstein had to satisfactorily explain a) how everone can see the same speed for light and b) what happens if you attempt to move faster than light.
Imagine, for a moment, I'm standing on a train. The train is moving at some speed. I shine a torch from one end of the train to the other. According to postulate 2, the light must leave the torch at the speed of light. Now, imagine you're standing on a nearby hill, watching this experiment. You see the train go by and the torch turn on. Before relativity, you'd expect to see the light leave the torch at a speed equal to the speed of light plus the speed of the train. But, by postulate 2, you see the light leave the torch at the speed of light. How can we both observe the same speed for light if we're both moving at different speeds?
This conundrum led Einstein to completely redefine the concepts of time, length and mass. It is this definition which gives rise to the oddities that Hawking is talking about. Essentially, what Einstein said was that, for the speed of light to be a universal constant, when something moves close to the speed of light space and time must change.
The general gist is that in order for the speed of light to remain constant, time must go slower for a person travelling quickly.