I've had this hanging around in my draft posts folder for some time. In one form or another probably more than a year, maybe even two at this point. I don't clearly remember the conversation that prompted it, but I thought that after such a long time it deserved to see the light of day. The explanation is very much simplified, but I think it makes for a reasonable explanation, and is based loosely on an old thought experiment I encountered at college.
When we talk about the speed of light we don't really mean the speed at which light travels. Light travels at different speeds depending on what material (or none) it is travelling through, which is part of why we see phenomena such as refraction and Cherenkov radiation. When we talk about the speed of light we really mean Einstein's proposed speed limit, which is the speed of massless particles (such as photons) travelling unimpeded through a vacuum. This is also the speed at which you would require an infinite amount of energy with which to accelerate any given mass to, something demonstrated by Einstein's most famous equation. The really difficult to grasp aspect of the speed of light is that it is a universal constant. The unimpeded speed of light in a vacuum is always the same, and this is true irrespective of your point of reference. That is to say that the unimpeded speed of light in a vacuum is not really subject to relativity in the way we might normally expect.
If you consider two cars travelling in the same direction, one travelling ahead at 20 mph and another at 10, we consider the speed of those cars relative to a common point of reference: the road. One covers 20 miles of road in an hour, the other 10. We can also consider their speeds relative to each other: the gap of road between them increases by 10 miles every hour because that's how much faster the lead car is travelling. We have an intrinsic understanding of this principle, going faster allows you to catch up if you're chasing, or never get caught if you are in front. The speed of light breaks this rule.
So let's swap the cars. We have a car in front going at 10 mph and a chasing car doing 20 mph. These speeds are relative to their common point of reference, again the road. We know that the chasing car is closing the gap by 10 miles per hour, except we're going to screw that up and apply the rules by which the speed of light plays. The chasing car's speed therefore is 20 miles an hour from the perspective of the road but must also be closing the gap on the first car by 20 miles per hour. The difference here is one of relativity, but things no longer add up. You can't really imagine a situation where the driver of the lead car sees the chasing car approach at 20 mph (we expect the speed difference to be 10 mph), but a pedestrian on the road side also sees the chase car moving at 20 mph. So what gives?
Speed is a simple calculation: distance over time. The distance isn't changing; whatever we define to be the length of the chase is the distance. Speed normally changes, but we have established that in this case it is a constant. So the only variable that can give in this scenario is time. Time must be different from the perspective of the bystander and from the perspective of the lead driver. This principle you know from the phrase "time dilation". The faster you go, the more time slows down from your frame of reference, and this is a difficult concept because it is wholly incongruent with our experience of the world.
This leads to interesting situations. If I am in a space ship travelling at light speed and look out of the rear window, what can I see? Well, any light travelling towards you, despite your travelling at light speed, is catching up to you at light speed. Principles like Doppler shift confuse things (and are beyond the scope of this little explanation), but the rear view would essentially be visible to you 'as normal'. It leads to the law of simultaneity, whereby a single event can appear to occur at a different time according to the frame of reference from which it is observed.