What is the function of the zener diodes used in this setup?
zener diodes (or diodes in general) only conducts current in one direction

That is only partly true, Zener diodes will also conduct when the reverse voltage exceeds a certain value. Here that value is 2.4 V (they are 2.4 V zeners). Add to that the forward voltage of the other Zener (about 0.7 V) and the two Zeners in series will start to conduct when the voltage across both exceeds 3.1 V. When that voltage is negative (-3.1 V) the same will happen.

These diodes limit the input voltage difference to the opamp. Too much voltage difference will destroy the opamp.