(TL,DR: see paragraph 5)
The meter module requires the voltage on its GND pin to be between its V+ and V- supply pins. It converts and displays the voltage between its IN+ pin and GND.
The 7810 regulates the input to 10V between nodes labelled +5V and -5V.
R2 and R3 provide a mid-point voltage with Thevenin impedance of 2.5 kOhm (= 10K // 10K) in parallel with 100nF. Thus any (DC) current drawn from or supplied to this node will push the voltage by 2.5V per milliamp.
The GND node will be carrying currents from: Meter VIN+, R9, R11, R15, R16, R13, A2 and C5. These will probably sum to less than a milliamp, but the meter may draw varying current through each measurement cycle.
Amplifier 1 acts as a voltage follower for the R2 R3 chain. It will act to hold its output, the node labelled GND, at the mid-point of nodes labelled +5V and -5V. Looking at it from a different perspective, it acts to pull the mid-point of its supplies towards its output voltage. It will have a closed-loop output impedance of a few ohms, so current drawn on the GND node will have little effect on the voltage between GND and the +5V and -5V lines.
Amplifiers II-IV are all configured as simple differential amplifiers. II and III have a gain of 100V/V and Zin of 10K. IV has a gain of 20 and Zin of 50K.
Having C5 connected directly to the output of amplifier IV is an error. OPAs are not specified to be stable with a large capacitative load. It would probably be better to put it across meter VIN+ and GND, with 10K or so between the wiper of A2 and the VIN+.
The gain of the circuit will directly depend upon the output voltage of the 7810. If the meter has an external reference input it would be best to connect this to a fraction of +5V, this giving a ratiometric reading.
The offset voltage of all four amplifers will contribute to the signal. The amplifiers will need good DC and 1/f noise specifications.