Why are audio transformers required?
This is a bit of a tricky circuit.

Normally, a transformer produces a voltage ratio matching its turns ratio, a current ratio that is the inverse of its turns ratio, and therefore an impedance ratio that is the square of its turns ratio.

Now in this circuit, we have a mystery box which is most likely a square wave clock oscillator.  By appearance, the transformer secondary is being used to couple the audio as an A/C "ripple" on top of it's power supply, in the hopes that this will produce some AM modulation of the output.

It's not entirely clear that the transformer is being correctly applied; without knowing the output impedance of what the jack is plugged into or the beyond-data-sheet properties of the oscillator, we can really only speculate if the transfer is best the way shown, turned around the other way, substituted with a 1:1, etc.  Likely this is a "pragmatic" circuit as much as a "calculated optimal" one.  

It's possible that the use of a transformer at all may be primarily to provide isolation between the circuits.  Powering through a small series resistor with a capacitor to couple in the audio could be another option, though perhaps less efficient.  

There are two additional problems which merit some thought before building this:

1) The oscillator probably isn't rated for a 9v supply.  Most want 5v, or 3.3 or perhaps today something even lower.  It's not clear that the DC resistance of the secondary will drop the supply voltage enough under this small load to be within the limits.

2) The oscillator is going to output a square wave, which is rich in harmonics.  Without a low pass filter to round the square wave to a perfect sine wave, this will not only transmit at 1 MHz as intended, but also at 3, 5, 7, 9, 11, etc MHz, potentially up into places where such spurious emission produces harmful interference (for example, 7 MHz + the audio frequency would land in the morse code allocation of the 40m ham band, where trying to receive extremely weak signals is common and interference detested) .  Needless to say, there are regulation about spectral purity for various transmitter power levels.