That device is actually incredibly simple, albeit slightly crude.
The circuit appears to be as follows... Part one is mains in, through a small switch (possibly mains rated, possibly not, but given the long shafted plastic button, not something I would worry about) which energizes a good old fashioned 1980s style wall wart transformer, from the days before even the cheapest wall wart had a switch mode power supply in it. I can't see the output side of the transformer, it may have a small resistor to load it up and increase the magnetic field, or it may simply be open circuit.
The magnetic field from that does the actual degausing.
Part two consists of a high value resistor (Brown Black Yellow = 100k Ohm) and LED (across the mains, slightly dubious practice, but not that dangerous), the LED only lights when the button is pushed.
One thing I would recommend is to ensure you have a 3A or even 1A fuse in the mains plug. From a safety perspective though, as cheap goods from China go, it looks reasonably safe. If the switch fails closed, the thing stays on (not a problem), if the switch fails open it fails to energizes and the light stays off.. also not a problem.
The only issue is that the unit has no built in fuse, so you would be relying on the plug fuse to blow if the transformer winding failed shorted, hence the comment about using a 3A or 1A fuse.
It certainly looks a lot less likely to burn the house down than a lot of badly constructed modern cheap wall warts, but I would still recommend unplugging it when not in use.
I might even get one. At round about a fiver, shipped from China, it is probably as easy and cheap to pick up one of these as it would be for me to go through my electronic scrap bin to scavenge the parts.