The Capacitor in the two resonant circuits makes each into a self-resonant device, capable of storing energy. The energy is stored in the magnetic field when the capacitor is discharging electric field, and in the capacitor when the coil's magnetic field is collapsing. This is different from a transformer because energy is not stored in an ideal transformer.
This design is almost exactly a
tesla coil. Tesla coils transfer energy from a tank circuit to a secondary circuit that share a common resonant frequency. The tank has a big capacitance and a small inductance, and the secondary has a tiny capacitance and a big inductance. The sparks are due to the voltage multiplication due to the ratio of turns from the tank inductance to the secondary. In tesla's design, a spark-gap is used on the tank side to initiate ringing by suddenly joining the tank L and C together in parallel through the low resistance plasma of the ark.
In the design on TED, the idea is the same. The turns ratio is different from a tesla coil since the purpose is to power devices that run on 120Vac or 5Vdc, and the power must be harvested in some way. I imagine that they insert a full wave rectifier and regulator for the phones, and an additional transformer coupled to the one for the TV that is rectified and powers an
inverter of some kind which makes 120V@60Hz. Modern electronics can make these conversions pretty efficient.
a guess: The fields are in the same direction because the two coils are magnetically coupled, combining to form a single resonant system.
A system like that would have harmonic resonances which would represent the two systems oscillating at different phase shifts with respect to each other. I imagine that the phase shift would be dependent on the amount of power transferred, maybe its 0° at zero power transfer and shifts to 180° at maximal power transfer. I can only assume that when the primary coil is energized for the first time, lenz's law will be obeyed and the reverse field will be initiated in the secondary, but as the energy of the secondary grows it becomes energetically favorable for the two fields to be in alignment. This could be where the energy to power the device comes from, the difference in
magnetic dipole potential between two aligned fields vs. two misaligned fields. But I don't know, http://ocw.mit.edu/courses/physics/8-03-physics-iii-vibrations-and-waves-fall-2004/video-lectures/lecture-6/.
Its possible that the artists rendering is just for illustrative purposes.