How does tuning in a spectrum analyzer actually work?
A RF mixer doesn't actually multiply two signals, at least not directly. Instead, the signals are first added and then passed through a nonlinear circuit, which ideally has a quadratic relationship between input and output.

We can therefore represent the mixer by a function:



And putting the sum of two signals into that function gives us:



This clearly contains the original input signals. Only the product of the two input signals is desired, the rest has to be filtered out.

Most mixers produce even more frequencies than those because they're not purely quadratic but also have summands with an even higher exponent. Third-order is especially nasty to filter out.

A typical mixer uses a diode as its nonlinear element, which has (roughly) the following function, ignoring some constant factors:



Computing its taylor series, we get:



This clearly has b != 0, so we get some of the original input signal at the mixer's output if we don't filter it out.