Can you tell me how the op-amp in this DMM circuit works?
Yes it is a confusing schematic.
THere are at least 3 poles on the rotary function switch which each have arrows to indicate the poles and the  position on the DC connector changes to connect to the current source on the lower side. Meanwhile there are reed relays with logical switch control names to show when it is closed.

The 8007 is used as a constant current source driven by IGSS of FET. The output drives the probes while the ADC senses the voltage drop across the probes using sense resistors of 1K, 10K, 100K, 1000K to generate a voltage proportional to reistance controlled by Reed Relays with under and overflow detection. The binary control of A,B,C,D relays determine the value of the ladder network and also used for gain control on AC,DC input voltage.