The mixer relies on the nonlinear I-V characteristic of the diodes to produce other frequencies (f1 + f2 and f1 -f2)... Since it's an unbalanced mixer, a part of f1 and f2 are also present at the output...
The low-pass filter removes f1, f2 and f1+f2 from the output and only passes f1-f2 (which is 1Khz in this case)