A model I encountered as a kid, back when the Space Race was in full swing, is still the simplest explanation I've found:
Think about an inflated balloon with its neck closed. It doesn't go anywhere, because the pressure in all directions is equal. (Which is what keeps it inflated, too.)
Now open the neck. What makes the balloon fly around isn't actually the air escaping through the neck -- it's the fact that there's no longer pressure backward against the balloon at that point, so the pressure on the opposite side is unbalanced and there's a net force pushing the balloon forward.
A rocket engine is, essentially, a rigid balloon with an open neck which continuously re-inflates itself. High pressure inside, a nozzle designed to exhaust as fast as possible minimizing pressure in the backward direction, unbalanced pressure forward... "Zoom!"
(Yes, I know this is an oversimplification. 326
But it's an explanation which is so straightforward that a third-grader, which I was at the time, could look at it and say "Of course, that's obvious!" Unfortunately my teacher at the time didn't take well to having her wrong explanation corrected by a third-grader, but that's a separate story.)
Late addition: This is also why Project Orion -- the ultimate low-tech nuclear rocket engine -- would have worked. Build a huge shield, mount your craft to it with shock absorbers, and set off an atom bomb on the far side. The part of the bomb's shock wave which hits the plate will drive the craft forward. The fact that the rest of the shock wave is completely uncontained is wasteful but doesn't keep this from working. You could do the same thing with smaller explosives, but then the inefficiency becomes a serious problem -- which is why an rocket engine has a reaction chamber and nozzle, to capture much more of that energy and use it as thrust.
(Calculating just how inefficient Orion would have been, how much thrust it would have produced anyway, and how to build a shield and shock absorbers which could handle this, is left as an exercise for the reader. Or you can look up the report of the team that first examined it. No joke.) 326