I suppose that, like almost every other explanation out there, this is going to come up a little short because it's trying to explain some seriously complex mathematics without using any of that maths whatsoever. Try explaining a musical piece without being able to listen to it or using any musical terms at all. Anyway, here's my attempt to explain what's going on.
The first key point is that gravity is actually a very weak force indeed. Stupidly weak. It's only when you have a whopping great amount of matter in one place that you start to notice it at all. This makes it anyway an incredibly hard force to study and understand, because its true nature is only really understandable in the most extreme conditions in the universe. The event that's been reported as observed in this experiment was two black holes, each with a mass about 30 times that of the sun, colliding and tearing each other apart, spinning really fast and finally merging in around a fifth of a second, before spitting out an energy equivalent to the mass of three Suns that sent ripples propagating through the universe at the speed of light, over a billion years ago. And the effect measured was something like a hair's width over the orbit of Neptune. Even if you don't quite understand what the effect is itself, the idea of being able to detect something that precise is in itself stupidly impressive. But anyway.
The picture to have in mind with gravity is the usual one thrown in at this point: space and time are a rubber sheet, and gravity is the effect caused when a mass rests on that sheet and distorts it. Far away from the mass spacetime is flat -- nearby, things are curved. The heavier the mass, the greater the curvature, etc etc. This analogy is really quite useful, particularly if you get a chance to set it up (eg use lots of fish-netting stretched out over some metal frame, put a basketball in the middle, and start trying to send ping-pong balls in straight lines; the paths these balls take are not going to be that far off from the paths a comet moving near to the sun would take).
The really interesting stuff happens when you start playing around with the large mass in the middle. Get it to move, rotate, orbit another large object really fast, spiral inwards. The shape of spacetime in this case starts to be affected in much more dramatic ways, essentially because it's no longer a static shape but one that will change over time. This seems intuitive enough, I think -- the object is moving, and if it's curving spacetime then the curves will move along with it in some sense. If in addition the motion is dramatic enough and rhythmic enough, then the resulting changes in shape will pulse and take on the form of waves, rippling through spacetime in all directions. As any such wave passes a given point, the shape of space (and the passage of time) would, briefly, be distorted. Space would stretch or squash, time would speed up a fraction and then slow down a fraction -- and then it would be back to normal an instant later (unless the source of the gravitational waves is long-lasting, in which case this would happen several times in a row).
Again, though, all of these effects are only really noticeable at the most extreme scales. I think it's probably true that, in principle, you can make gravitational waves yourself (by, eg, spinning around on a chair), but the characteristic size of the resulting waves would be probably equivalent not to a hair in the solar system, but to (at most) an atomic nucleus in the visible universe. These really are tiny effects. It is little wonder, then, that Einstein, when he predicted the existence of waves, noted despondently that he would not expect to see them in any experiments. I'm sure he would be thrilled to have been wrong about that.
I've not read this explanation back yet -- I hope it's fairly clear, but if not or if you have any other questions let me know.