This water-saving concept is so simple yet so alien that it's hard to wrap the mind around it. If you live in an area where water is an issue (and we'll all eventually be there, I imagine), then it might be worth your while to puzzle through this concept.
So what happens when you flush the toilet? The water in the tank and bowl, along with the forces of gravity, combine to flush the waste away. Then, fresh water flows in to eventually fill these two vessels.
Now, what if the fresh water flowing in was diverted before it went into the tank and bowl? What if that fresh water came out a little spigot on top of the lid and became a little hand-rinsing station? Then, that water would drain into the bowl, ready for the next flush, and be officially called gray water, and that's what you'd be using for your next flush instead of fresh water.
The way it works, the fresh water flowing into the sink only runs as long as the tank is being filled after each flush. And that, of course, is when you'd want to wash your hands anyway, right after a flush. The downside is that you have to bend over the toilet seat to use this device. And that could be a little awkward.