I haven't given it much more than 1 second of thought here, but when there IS power, the AC leads coming into the light bulb will have a voltage, so the light knows that AC is on. It will use the AC power from the grid to power the LED and charge the battery inside.
When the switch is turned off (or a blackout occurs), the LED will see the AC power go off, and it will not know if it is a real blackout (no power) or the user just turned off the switch and WANTS the light off. To determine which case it is, the light bulb electronics will need to inject a small (possibly AC?) current into one side of the AC line and see if it can detect that current on the other line side. If it can detect the current on the other side of the line, then the AC power is off (blackout) but the switch is ON (closed). So the LED will come on. If it cannot detect that current, then the switch is OFF (blackout or not, doesn't matter, the LED light will stay off). It probably does this check every couple hundred milliseconds or so while the AC power is OFF (pulse check)
It can do this because the 2 wires from the light bulb through the switch are connected via the transformer at the house or the pole/utility box. So it can tell if the switch is open or closed. See the diagram below.
Pretty ingenious, actually. And could indeed be one of the few successful kickstarters.
(noted on their kickstarter page, it doesn't work with multiple bulbs on the same circuit. They don't explain why, but given my supposition above, I suspect that it cannot detect the switch open/closed position because the filament of the other bulbs on the same circuit interfere and complete the circuit ahead of the switch -- second drawing below)
![](http://i.imgur.com/0DNIhvpm.png)