Why A product detector is just a mixer: it multiplies the incoming IF signal by a locally generated carrier from a beat frequency oscillator (BFO), and the difference frequency that falls out is the recovered audio. SSB and CW signals have no carrier of their own, so the BFO supplies the missing carrier that the detector needs to demodulate them. A simple envelope (diode) detector cannot do this, which is why SSB and CW receivers use product detectors.
Watch out The choice about detecting spurious mixing products is a play on the word 'product' in the name; the 'product' here refers to the mathematical multiplication of two signals, not to unwanted mixer outputs. FM receivers use discriminators or ratio detectors, not product detectors.
Product = multiply: IF times BFO equals audio. It is the SSB/CW demodulator.