The emitter—we're using the vocab and italics from the original paper because they are awesome—wears an EEG cap on her scalp that records the electrical activity in her brain. The cap communicates wirelessly with a laptop that shows, on its screen, a white circle on a black background.
The emitter translates the message she wants to send into an obscure five-bit binary system called Bacon's cipher, which is more compact than the binary code that computers use.
The emitter now has to enter that binary string into the laptop using her thoughts. She does this by using her thoughts to move the white circle on-screen to different corners of the screen. (Upper right corner for "1," bottom right corner for "0.") This part of the process takes advantage of technology that several labs have developed, to allow people with paralysis to control computer cursors or robot arms.
The emitter's binary message gets sent over the Internet, yay.
The receivers sit inside a transcranial magnetic stimulation machine that's able to send electromagnetic pulses through people's skulls. The pulses make the receivers see flashes of light in their peripheral vision that aren't actually there. In addition, the machine has a robotic arm that's able to aim at different places on the receivers' skulls. The results are phantom flashes (called phosphenes) that seem to show up in different positions in the air, which is not spooky at all, no.
As soon as the receivers' machine gets the emitter's binary message over the Internet, the machine gets to work. It moves its robotic arm around, sending phosphenes to the receivers at different positions on their skulls. Flashes appearing in one position correspond to 1s in the emitter's message, while flashes appearing in another position correspond to 0s.
We don't know how the receivers keep track of all that flashing. Perhaps they take notes using a pen and paper.