What makes RF and LO harmonics?

It's due to distortion, that is, nonlinear response of the circuit.

A sine wave applied to a nonlinear circuit causes harmonics to be generated. that is, the output is not a perfect sine wave.

Fourier analyisis is a way to analyze those harmonics, it's not the cause.

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Harmonics are generally generated by non linearities in system, there are components that have a wanted non linear behaviour such as mixers and linear circuits which are getting into non linear operating regions such as amplifiers (unwanted behavior)
The harmonics of a LO are usually caused because of the device operating non linear, they can be minimized by proper biasing of the active device as well as filtering.