If you terminate your current transformer with a resistor to get a voltage signal out, a normal AC current transformer would saturate very quickly if you apply any DC current due to Vt=NAB where V is positive value (or negative, same effect) and as t goes on, B just keeps building until saturation. For AC, V keeps changing and nets out to zero.
In a zero flux current transformer, you use an amplifier to sense your output and drive a current in a secondary coil, wound in the opposite direction (or apply an inverse signal, take your pick) to cancel out the flux build up. This results in a new operating point where equilibrium is reached and the core won't saturate yet you can measure the current as a DC voltage which is proportional to your applied current in the secondary coil.
Here is an image borrowed from Hioki, a manifacturer of such modules: