Mystery of the Slingshot Maneuver

Originally posted by Astyanax
However, its direction of motion and hence its velocity relative to the Sun have altered. That's really all there is to it.

Here is a more detailed explanation from Wikipedia. You'll find the inset diagram helps a lot.

It's not just the DIRECTION of the motion that's changing. In the reference frame of the Sun the magnitude of the velocity will change as well (as your link correctly explains).

Imagine that you roll a basketball at high speed, and bounce a tennis ball off it. The tennis ball may be moving faster after the collision, in the lab frame of reference.