A frequency mixer is another way to say multiplier. When two frequencies, are mixed or multiplied, the result is the sum and difference of the frequencies. This all goes back to the trig identity:
sin(x) sin(y) = (cos(x-y) - cos(x+y)) / 2
which says that when you multiply two frequencies together, the resulting output is the sum and difference of the two frequencies. This is important for demodulation of AM signals. One of the two input signals is the modulated signal, whilst the other is the carrier. Therefore, the output signal from the identity is the carrier minus the modulated signal, which is the intelligence signal (what was modulated at the transmitter, i.e. what you want to listen to), and a much higher representation, which is filtered out. If this sounds a bit cryptic, research the demodulation of AM.