Why A mixer is a nonlinear (multiplying) stage, and multiplying two sine waves at f1 and f2 produces new products at f1+f2 and f1-f2. Some of each original input also leaks through to the output, so the principal components you see are f1, f2, f1+f2 and f1-f2. A filter after the mixer then selects the one product you want, which is why a 14.2 MHz signal mixed with a 5.745 MHz LO can be converted to a 8.455 MHz IF (the difference) or 19.945 MHz (the sum).
Watch out Doubling and quadrupling the input frequency is what a frequency multiplier or a harmonic generator does, not a mixer; the 1.414 and 0.707 numbers come from peak/RMS voltage relationships and have nothing to do with mixing products.
Mixers add and subtract: sum and difference, plus the two originals leaking through.