Nonlinearity by itself does not directly result in frequency conversion. Rather, the characteristics of the circuit are modulated by the presence of a large unrelated signal such as an LO if the circuit is nonlinear. Thus, if the large signal is periodic, it acts to periodically modulate the characteristics of the circuit, characteristics like effective gain. Thus, to the information signal, which we will assume is small, the circuit appears as if it is a periodically time-varying linear circuit, and that is what causes the frequency conversion.

-Ken