If he is just using low frequency signal, like AC from the wall, then you do not really need to worry too much about antenna shape and design. Anything you make of reasonable size will be a "small" antenna. Steven Best has done a lot of papers on small wire antennas and the last few papers I read seemed to confirm that the radiation properties were fairly independent of the wire antenna form, it mainly changed the impedance characteristics and efficiency.
So just loops would work fine. It appears though that the "figure 8" loop is probably two opposing loop antennas. A loop antenna is an approximation of a magnetic dipole antenna, so only one loop (or a coil of loops) would be fine. The number of coils and increasing the area of the loop would improve the radiation properties. However, he states in the description that the figure 8 is meant to confine the radiated field. If we place two dipoles close to each other and operate them in opposite modes, the fields in the far-field will quickly die off. So it is my guess that he is simply looking to run two loop antennas whose currents run in opposite directions, the result being that the fields are very localized. You could do this by running the wire in a figure 8, the cross point causes the direction of the current flow to switch. A similar idea is used in a folded wire dipole of one wavelength in length (except it is meant to improve the radiation and not stiffle it). Just make sure to insulate the cross-point, it is not meant as a short between the wires.
The receiving antenna uses much smaller loops, so he has boosted the gain by winding many loops and inserting an iron core. As to what voltage on the transmitting antenna, *shrug*. Who knows, it all depends on how strong of a signal you need to receive, I would just be reasonable about it. You do need to have an AC signal, you might be able to just take the signal off of the house mains but step it down in voltage. Something on the order of a volt or so would be a starting point. A typical LED only needs 0.7 V to turn on, so it would be a matter of slowly ramping up the input voltage to your transmitting antenna until you met that requirement but I do not think you would want to overdo it.
EDIT: There is a better video:
http://www.youtube.com/watch?v=SSgo_N-5JOg&NR=1
You can see that he is working in the low MHz range and you can more easily see that he has added a capacitor and an inductor (I guess the wire coil can provide the necessary inductance) as a tuning circuit. What I mentioned above, just making loops and pulling the AC off of mains could work but it would be very very VERY inefficient and may not be able to work in actual practice. Both videos probably used MHz signals and matching networks to achieve a high enough received signal to power the LEDs.