If the gravitational attractor (say, the Sun) is just sitting there minding its own business, then yes, the gravitational field ("dent" in the space-time rubber sheet) is constant and any other object (e.g. a comet) which wanders by will "feel" the effect of the gravitational curvature "immediately".
But what this discovery describes is how quickly the "rubber sheet" responds to *changes*. For example, if you just plunked the Sun down into a spot of space where it didn't previously reside, the question is how long it would take the resulting "curvature" of the "rubber sheet" to propagate outward. Or alternately, if you start rolling the ball (Sun) north across the sheet, does the rubber sheet instantly adjust as the ball rolls, or does it take a little time for it to "catch up" (i.e., will a spot of curvature 50 yards away instantly feel the change, or will the changing curvature have to "ripple" out there like a crowd doing The Wave?
This is hard to describe will in words, an animation would be ideal, but I don't know of any on the web which show this.
Gotcha.