Can you receive a shock by touching only one probe ?
He still touches button with its contacts by the other hand. That button is residing in the low voltage circuit, which is connected to the ground. Circuit diagram is at 3:28 of the video.

And finally, arc is going to find the "shortest" way to get through to the ground; the resistance of the body is much less than resistance of the air, even if it looks like physical distance between contacts is much less than distance between his fingers of the right and left hand. So it is fairly logical that it diverted to him rather than continuing going through the air between the contacts.