Understanding a simple crystal oscillator transmitter
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.