In near field zone (low frequences, mains and DC-DC converters) the device acts like a secondary coil of a transformer, so it drains power from some other device.

In far field zone (high frequences, bluetooth, wifi) it captures part of already emmited wave, so it doesn't place additional load on emitter directly. But since device blocks wave propagation through itself if placed between transmitter and receiver it will degrade reception. And that in turn can result in automatic increase of transmitter power (in case of GSM phones at least). Which will increase transmitter consumption.

Think of it as a tremendously-inefficiently-magnetically-coupled transformer. The power line or wifi antenna is the "primary winding" and the antenna in this box is the "secondary winding" and the air in between them is doing an awful job of being the iron plates a normal transformer winds those wires around to direct and concentrate the magnetic fields. Power drawn out of the secondary coil results in power draw in te primary coil (multiplied by the inefficiency of the transformer). The reason this appears to work is that no-ones measuring the increased power draw from the source - since the device is only drawing microamps or possibly single digit milliamps at a volt or two, no power company is going to notice a few milliwatts of extra load.

On the other hand, if you stacked 20 or 30 of these around your ~100mW wifi basestation, where do you suppose the power extracted comes from (and, as you suggest,what do you suppose would happen to your wifi range? I also suspect loading up the transmitting antenna like that will change enough about the way the transmitter/antenna system works, and degrade the transmitter output in even more serious ways than just the raw power extracted might indicate - I'l guess the frequency and resonance assumptions made by the transmitter designer would all be wrong.)

I don't know why the parent was downvoted, it's an honest question that's encouraged lots of high-quality responses. Here's mine:

Faraday's law of induction: A changing electric field through a coil induces a current in that coil. Induction is how transformers work. Hopefully you believe that attaching things to a transformer's output can put a load on its input.

Since this device and transformers both use induction, if it can happen in a transformer, it can happen with this technology.

Every bit of metal acts like an antenna. You could think of it as the receiver creating it's own signal that counteracts the transmitter. (Powered, of course, by the transmitter itself.)