I found that an explanation based on
control systems helped me the most. And since people use
oscilloscopes to measure this stuff anyway, it's probably pretty close to the truth.
When you take the position of your particle, it looks like an impulse. When you take the velocity, it looks like a step function.
The problem is that while a pure impulse is accurate in position, its area is indeterminate. And where a step function tells you something about magnitude, it is homogeneous over time, and cannot be pinned down anywhere.
Whenever we measure a particle, we always (by definition) use some sort of filter that gives us the position and velocity of the particle to some degree, and these degrees of error are inherently dependent on each other.
Another way to see it: Think of a photograph of a bullet in flight. We can take a picture with a normal camera, and see a streak on the film where it passes. We can't easily measure its position, but by noting the time of exposure and measuring the streak, we can tell how fast it's going. If we take a picture with a high-speed device, we'll get a clear picture of the bullet's position, but no indication of how fast it's going. If there is streaking, then the speed measurement will be much much less accurate than had the picture been taken at a slower speed, which gives a better sample. Our 'interference through measurement' that the romantics like so much is never even an issue.