I think I figured it out thanks to Oli Charlesworth's comment 'If you draw a diagram of what is happening, you will get a triangle! ;)' and a fair amount of diagrams. I'll try to explain it, but I've never had math in English (I'm Dutch), so I don't know the appropriate mathematical terms for everything, but I'll try:
I first didn't understand where O. Charlesworth's triangle was. But after a few drawings I figured out that the line of the movement the 'player' makes forms a triangle with the x axis or the y axis.
After that I still didn't understand what sine and cosine did there, but after rereading and generalising my math book I figured out that the the relation between the x-axis and the hypothenuse (the movement of the player - hey, I knew the English word for it!) equaled the distance along the x-axis traveled by the 'player'. Fortunately (but not coincidentally, it's MATHS), this is exactly what the cosine returns. If you replace x-axis with y-axis and cosine with sine, you got my explanation of why ySpeed = ...
works.
I fear my English explanation is not entirely correct, but that's because of my English skills, not my maths skills. I hope it helps someone.
PS I now realize my question was really vague. It's a miracle it isn't already closed.