How the diode in this circuit achieves AM modulation
The diode in this circuit does not "achieve modulation". It's functioning as a "clamp" or "DC restoration" function so that the varying amplitude of the video does not affect the overall brighness in an objectionable way.

It serves to keep the tips of the sync pulses (the most negative voltage within the composite video signal, corresponding to maximum RF amplitude) at a fixed voltage value. The actual modulation occurs in the BE junction of the transistor.