Why A balanced modulator is a mixer whose circuit balance cancels the carrier, so only the sum and difference products of the audio and the carrier oscillator appear at the output. That gives double-sideband suppressed-carrier RF, the first stage in generating SSB. A filter or phasing network then removes one sideband to leave single sideband.
Watch out The audio answers describe detectors or audio processors, not a modulator; a balanced modulator takes audio in and puts RF out. Frequency modulation is produced by a reactance modulator or a phase modulator, not by a balanced mixer.
Balanced means the carrier is balanced out: what's left is both sidebands, DSB suppressed carrier.