Rotation Equations've Got Me Stumped.

I perfectly understand the simple yet inefficient rotation equations. Like to rotate on the z axis:

x=cos(°)*r

y=sin(°)*r

That's easy and all, but inefficient. If you rotate 89° on the z axis, when you then go to rotate on the y-axis, you have to re-calculate the radius again(which would end up being WAY smaller). But I don't get these equations which don't require any radius length for rotation:

x=cos(°)*x' - sin(°)*y'

y=sin(°)*x' + cos(°)*y'

As much as I've tried, I just don't get it. I fail to see how it works. Can someone please walk me through this? You might have to talk to me like I'm a 3-year-old for me to get it. :-\