How does a Gilbert cell work?
The six-transistor 'multiplier' (or modulator, or demodulator) circuit
has continuous output dependent on two differential-mode input signals.
Call the two inputs A (the Q5 and Q6 base voltages) and B (Q1 and Q2 bases).
Then the differential output (Q1 and Q2 collectors) can be expressed (because it is a continuous, smooth
function of the inputs) as
$$Vout = C_{00} + C_{10} A + C_{01} B + C_{20} A^2 + C_{11} A B + C_{22} B^2 + ...$$
This is the Taylor series expansion, with higher order terms omitted.
Note, by symmetry, $$C_{00}= C_{01} = C_{10} = C_{20} = C_{02} = 0$$ 
This means that the lowest-order nonvanishing term is the product A X B.
So, for small signals (which allows us to ignore the higher terms), the
circuit is an analog multiplier of the A and B signals.