In this circuit i know that the diode mixes the signal. I know that after the diode it is a half-wave. I know that the capacitor reproduces the lost signal. I have a feeling L2 is maybe plays a part in this signal regeneration?
The carrier and audio modulating signals are simply added but do not form a modulated signal.
The diode rectifies this signal forming a crudely modulated signal which contains a DC component, low frequency component and high frequency component.
The inductor acts as a low impedance for the low frequency component and high impedance for the high frequency.
The capacitor blocks the DC component but passes the high frequency AC signal
The tuned LC circuit filters out all but the the desired AM signal.
The diode load completes the path for DC return. It could be a resistor also. If not present the input signal would charge the capacitor to the maximum value and than the whole process of mixing would stop.
You want to develop the modulated RF signal as a voltage across an impedance, but you don't really want the baseband audio signal there. So the best choice of load there is a high impedance at RF frequencies, but low at DC and audio frequencies - hence, an inductor.
The diode is essentially a thresholding device here. The instantaneous voltage of the carrier plus modulation signal has to be above some voltage to make it thru the diode.
Think of the modulation signal as slowly varying DC. When the level is low, most of the carrier will be cut off. Think of the limiting case where the DC is substantially negative. The diode will always be reverse-biased and nothing makes it thru to the inductor. The DC level sets where in the carrier waveform the cutoff will be.
The average of the modulation signal is adjusted so that half the carrier is clipped off. The modulation signal can then be up to the same amplitude as the carrier. At the top end, all the carrier is passed, and at the bottom end none of it.
The result is a cut off sine wave. That's not the same as a amplitude-reduced sine wave. The cut off sine has a lot of harmonics, with the desired amplitude-modulated sine being in the mix. The remaining inductors and capacitors are set to resonate at the carrier frequency. This filters out all the components of the messy cut-sine signal except those close to the carrier frequency. The result is then a amplitude-modulated sine, as intended.
Note that this kind of primitive AM modulator is not all that linear. It's good enough to be useful for voice, for example, but quality AM transmitters use more sophisticated mixers.
I just realized the question is actually about the inductor, L2, not the diode. As it says, the diode needs some impedance to work against. A resistor could have worked too. However, a fairly high Q filter must still be applied to the clipped-sine signal. L2, together with the other inductors and capacitors form this filter. L2 therefore serves two purposes, to provide a load to the diode, and as part of the filter that attenuates all the unwanted crap in the clipped-sine signal.