It looks like simple ratio gears in series. Each alternating dial turns in the reverse direction of the previous. Since all you're doing is reading off numbers, the direction of rotation doesn't matter.

If you watch it for a bit, one dial will move. It's probably the one on the far right, but you can verify this through observtion. The dial on the opposite end probably won't move perceptively as long as you live there - that's a total number since the device was installed and it could take decades to increment by one.

Take a picture every month on the same month-day; make a spreadsheet and record your kwh cost, and multiply that by the difference from the reading from the previous month. Taxes and fees aside, that should be your bill.

How you fight this overcharging & occasional refund, I have no idea. You might call your local township/city/whatever and see if you can track down the regulation office and find out if you can legally file a complaint, or if this practice is allowed.