Oh my.. that's really nasty.. but it works
The oscillator is this:
http://www.foxonline.com/pdfs/f1100e.pdfIt outputs a TTL square wave and is meant to run on 4.5-5.5 volts. Here he uses more like 6 volts (depends on how used the batteries are), but it appears to handle that just fine.
To perform the amplitude modulation, the transformer is connected in series with the supply voltage, so it can add and subtract the amplified audio signal. If we say the output of the transformer is +/- 1 volts peak, then the oscillator will run on a varying 6 +/-1 volts supply (5 to 7) and the amplitude of the square wave will change accordingly.
This is abuse of a poor clock chip for a digital system, but it very roughly does what you need for an AM transmitter.
I didn't watch the second video yet, but you can use fixed components. Not sure why you would do that however, as you would have very little idea of what it would be tuned to, and thus little chance of picking up your transmitted signal.