It mostly works because the object you're sling-shotting with is moving relative to the initial origin of the probe - if you can line it up so the object is moving in the direction you want to accelerate in you can get it to "drag" your probe along a bit.

The end result is the probe isn't moving any "faster" leaving the slingshot object than it arrived from that object's point of view, but it may well be faster from the origin reference frame.

The thing is to remember that the bodies you're launching from, and then slingshotting against, are moving very fast relative to each other already. It's like throwing a bouncy-ball against a moving car - when it bounces off it may be moving faster relative to you than when you threw it, but it's actually a lower velocity relative to the moving car's reference frame.