How can over 24 GHz communication be possible?
RF comms do not transmit one bit of information per cycle of the carrier wave - that would be digital baseband communications and it requires incredible amounts of bandwidth.  Incidentally, you can buy FPGAs with built-in 28 Gbps serdes hard blocks.  These can serialize and deserialize data for 100G ethernet (4x25G + coding overhead).  I suppose the 'fundamental' frequency in this case would actually be 14 GHz (data rate/2 - think about why this is!) and they require around 200 MHz to 14 GHz of bandwidth.  They don't go all the way down to DC due to using the 64b66b line code.  The frequency used to drive the serdes modules would be generated by some sort of a VCO that is phase locked to a crystal reference oscillator.  

In the RF world, the message signal is modulated onto a carrier which is then upconverted to the required frequency for transmission with mixers.  These balloons probably have a baseband of less than 100 MHz, meaning that initially the digital data is modulated onto a relatively low frequency carrier (intermediate frequency) of around 100 MHz.  This modulation can be done digitally and the modulated IF generated by a high speed DAC.  Then this frequency is translated up to 24 GHz with a 23.9 GHz oscillator and a mixer.  The resulting signal will extend from 23.95 to 24.05 GHz, 100 MHz of bandwidth.

There are many ways to build high frequency oscillators in that band.  One method is to build a DRO, which is a dielectric resonance oscillator.  Think of this as an LC tank circuit - there will be some frequency where it will 'resonate' and either generate a very high or very low impedance.  You could also think of this as a narrow bandpass filter.  In a DRO, a piece of dielectric is used - usually some sort of ceramic, I believe - that resonates at the frequency of interest.  The physical size and shape determine the frequency.  All you need to do to turn it into a frequency source is add some gain.  There are also ways of using special diodes that exhibit negative resistance.  A Gunn diode is one example.  Biasing a Gunn diode correctly will cause it to oscillate at several GHz.  Another possibility is something called a YIG oscillator.  YIG stands for Yttrium Iron Garnet.  It is common to build bandpass filters by taking a small YIG sphere and coupling it to a pair of transmission lines.  YIG happens to be sensitive to magnetic fields, so you can tune or sweep the center frequency of the filter by varying the ambient magnetic field.  Add an amplifier, and you have a tunable oscillator.  It's relatively easy to put a YIG in a PLL.  The power of a YIG is that it is possible to use it to produce a very wide band smooth sweep, and hence they are often used in RF test equipment such as spectrum and network analyzers and sweeping and CW RF sources.  Another method is to simply use a bunch of frequency multipliers.  Any nonlinear element (such as a diode) will produce frequency components at multiples of the input frequency (2x, 3x, 4x, 5x, etc.).  Stringing together a chain of multipliers, bandpass filters, and amplifiers can be used to produce very high frequencies.