What on earth is going on?
The answer is that we need to reconsider our notion of time. Because the speed of light stays the same, observers moving with respect to one another don't just have different ideas of 'still', 'moving', and 'how fast': they also have different ideas of 'now', 'simultaneous' and 'how fast time is moving'. Most of the time we can ignore this difference because the speed of light is enormous compared to the speed of the average train and any difference in timing is negligible. (For a train going at 100 miles per hour and a platform 100 metres long, in the time it takes the light to reach the end of the platform the train will have travelled less than 1 hundredth of a millimetre.) Once we're talking about some of the fundamental particles the sun throws out into the solar system, though, the difference made by relativity becomes quite dramatic.
The theory of special (that is, constant velocity) relativity follows from these simple ideas. Hopefully I'll find the time to explain how - the only maths you'll need is Pythagoras' Thoerem.
In the meantime, click here to see things from the platform again. Or just return to main menu.