Flying probes are used for testing printed circuit boards (PCBs) and the chips mounted on the PCBs. One or more probes touch various parts of the circuit board and send an electrical signal into the circuit and a different probe checks to see if the signal at some other part of the circuit successfully matches what is expected.
I'd never seen them in action but then I discovered that there are several videos on Youtube. They are incredibly fast! The good stuff starts at 0:15, after the machine has done a computer-vision registration of the board. I don't quite understand what the blunt piston-looking thing is doing though.