I try to understand the AM transmitter on the figure below, but I cannot figure out two essential things. The carrier frequency is created by the crystal oscillator chip. 
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Why does it suffice to put the audio input in parallel to the battery, in order to add this signal (baseband signal) to the carrier frequency?
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Why do we need the capacitor?
Best Answer
This is a very crappy "AM transmitter" circuit which isn't worth its name.
It "works" by modulating the supply voltage with an Audio signal.
The output of a crystal oscillator switches between negative and positive supply. By changing that supply with a signal, the amplitude of the oscillator's output signal will change with that signal.
The "modulation depth" of this circuit will be bad as the crystal will only work for a certain voltage range limiting the amount of AM modulation possible.
It would make more sense to have the 470 ohm resistor in the Bat +5V line instead of the Bat - line as then ground of the audio circuit and this circuit would be the same.
The capacitor is needed to prevent +5 V getting into the audio circuit, without the capacitor the audio circuit might be damaged if it does not have a DC-blocking capacitor of its own.
As I said, this is a really crappy circuit and although it will do "something" it is more a "hack" than a well designed circuit. So don't expect too much from it.