The fact that theremins use heterodyne mixers has nothing to do with RF. The 'antennae' are not antennae in the classical, RF sense. The capacitance explanation is correct.
Capacitors and Theremin 'Antennae'
The simplest type of capacitor is a parallel-plate capacitor. That means the capacitor consists of two metal plates separated by some material called the dielectric. The equation for the capacitance of such a capacitor is C=εA/d, where ε is the permittivity of the dielectric (ε≈8.8541878176..×10^−12 F/m for air).
When you are operating a theremin, your hand is one plate (your hand is effectively grounded), the antenna is the other, and the air between the two is the dielectric. As you move your hand, you vary the capacitance between ground and the antenna. Both hands will affect both antennae, as they act like two plates in parallel, increasing the total area.
The two antennae are at right angles because that reduces the impact your left hand will have on the right antenna and vice versa. For example, as you move your hand up and down above the volume antenna, it maintains a relatively constant distance from the pitch antenna, thus it's contribution to the overall capacitance is constant (and small).
Theory of Operation
Note/Update: Please refer to FredM's Answer for a more detailed description of the oscillator.
Both antennae capacitors are part of two different, complex active LC oscillators. The 'L' refers to inductors, which store energy in a magnetic field; the 'C' refers to capacitors, which store energy in an electric field. In an LC oscillator, energy is constantly flowing back and forth between the two, changing from electric potential to magnetic potential.
The frequency of the pitch oscillator is beyond audio frequencies, so it can't be directly used. The theremin has a third oscillator that operates at a fixed frequency. The pitch oscillator and the fixed oscillator's outputs are fed into a heterodyne mixer, resulting in an output that includes the sum and difference frequencies of the two inputs. The sum frequency is even higher than the original signal, thus it is useless and is filtered out. The resulting signal is a single frequency (plus harmonics) in the audio range.
The frequency of the volume oscillator is used to control how much the audio signal is amplified. As you move your hand, the frequency changes, so the amplifier's gain changes, and thus the output volume changes.
The big question is: what distance do you want to cover? The data sheet of the transmitter quotes a maximum range of 50 metres [about 150 ft]. Will you use that, or will the receiver be closer?
Any oscillating signal will radiate: the whole point of the USA's FCC is to limit the amount of annoying [or dangerous] EM radiation coming from devices. Depending on the range, and the presence of bulky metallic objects between transmit and receive, Your Mileage May Vary.
Antenna theory and design can be taught in depth by amateur radio enthusiasts, or groups like ARRL. For starters, a simple piece of wire about 20cm long can act like a "whip" antenna: keep it clear of grounded cases, and it should be enough to get you started. A second piece the same length can act as your receive antenna. Start with the circuits next to each other, make sure they work, then seperate them. If they stop working before reaching the seperation you want, THEN [and only then!] consider delving into antenna theory...
If you need more, I'd suggest TI's Application Note #AN058: http://www.ti.com/lit/an/swra161b/swra161b.pdf
And: http://www.picaxe.orconhosting.net.nz/yagi433.jpg
Best Answer
A monopole antenna (for example) at exactly one quarter wavelength will project an impedance of about 37 ohms resistive and about 20 ohms reactive (inductive). At longer lengths it will project more resistance and more inductive reactance and close to 0.47 wavelengths it will be highly resistive and have neglible reactance as per the diagram below: -
So, if you want to tune a long antenna with capacitance or a short antenna with inductance that's fine but be aware of how the resitance (a projection of the resistance of free space) will change significantly and how the cycle repeats itself at 3/4 of a wavelength.
It's somewhat incorrect to think this way - all the inductor does is counter the inherent increase in capacitive reactance (by series tuning) thus leaving just the resistive component (and this will be very much smaller than the resistive component at a more appropriate length).
The impact of this is that the signal received is much smaller so, it's not really lengthening the antenna at all; it's tuning out an impedance that could block the now much smaller signal being received.
If you take that into account then certainly tuning a long antenna is fine but somewhat pointless in many applications where resorting to an antenna of about the right length will yield much more favourable results.