DSB-SC modulation circuit
MC1496 is an imperfect design and when used as a regular AM type broadcast modulator then there is no need for the fine-tuning of the DC offsets at the audio input. This is because regular AM broadcasts modulate less than 100%. This can be achieved by using a 4 quadrant modulator (like the 1496) and adding a dc offset to the audio input so in fact, any small error in dc offset introduced by the chip doesn't amount to a big deal performance wise. This turns it from 4-quadrant multiplication to 2-quadrant multiplication: -



However, when it comes to double-sideband-suppressed-carrier (aka modulating greater than 100%) to truly suppress the carrier, the audio input needs to have its dc component removed and, the dc imperfections in the 1496 need balancing-out. Below is what happen when you over-modulate a little bit, leaving some carrier behind but going beyond the AM broadcast levels of modulation: -



A regular AM receiver will produce distortion (as per the top-right diagram) due to the modulation being greater than 100%. Full-fat DSBSC looks like this and a regular AM detector would produce Dalek type sounds instead of pleasant normal audio: -



And to truly 99.9% suppress the carrier (i.e. not have hardly any spectral content whatsoever) you have to fine tune the dc offsets so that they virtually disappear.

Removal of DC on the carrier input as well as the audio input is equally needed because, for the simple fact being that the output = \$ Audio\times Carrier\$ i.e. just straightforward multiplication in the time domain.

EDIT

The op's circuit indicates that the audio is fed thru a 0.1uF capacitor which then feeds a 300 ohm resistor to ground. This cannot be correct if we are to believe the label "audio signal input" because the high-pass cut-off frequency would be \$\frac{1}{2\pi RC}\$ = 5305 Hz and this excludes most of the relevant part of the audio range. Here is another similar 1496 diagram for reference: -



Note that this uses a 10uF capacitor and a 100 ohm resistor giving a cut-off of 159 Hz - not hi-fi but good enough for speach.