If one flips the rubbersheet, the geodesics are the same, but clearly now the model doesn't match our intuition. Indeed the objects are attracted by the mass because of the Earth's gravitational potential, not because of the rubbersheet curvature. The balls are following the variation in height of the sheet (in a sense a first order variation) not the curvature of it (the second order variation). The model is bending the wrong component (space, instead of time), and with the wrong sign (the geodesics are repelled by the mass).
As pointed out by Ron Maimon in this answer: http://physics.stackexchange.com/questions/1019/common-false...
a much better explanation can be offered introducing proper time and the effect of gravity on it.
Could you explain it to me this way?
Finally, an explanation of gravity in layman's terms.
Still, there is the kernel of the correct explanation which could probably be presented to the larger public, starting with the well known explanation for refraction in terms of minimal paths (the lifeguard trying to save a drowning man), passing to the fact that in special relativity the trajectory of the free particle maximizes the proper time, and ending up with the gravitational case.
It's not hard to summarize the basic tenants of GR, here is a very good attempt: http://physics.stackexchange.com/a/3018/852. For a layperson I would focus on (1), what a geodesic means (what's the shortest way to travel from NYC to London, etc.). Then I would move to (2) and say that we don't know why that is the case. This would be a nice segue into a discussion about how Physics just describes observed phenomena with equations and the explanations are iterative.