If a spacecraft uses a massive object for a "slingshot", it is passing through more "space". Since the massive object "warps" the space (compresses it) then the spacecraft covers more distance in the same time as a similar spacecraft that doesn't pass close to a massive object, therefore the slingshot spacecraft is "accelerated" from the viewpoint of the second spacecraft. Instead of the standard "rubber sheet" with bowling balls, I think of the warping of space around a massive object (or ANY object with ANY mass) as a pane of broken glass with a, say, bullet hole representing the mass in space... Still very two dimensional, but maybe it works for you, too!
