There are several ways to install an inverter with a transfer switch. The incoming shorepower first goes to a double pole breaker - your main breaker. One of the breakers on the panel feeds the inverter AC input. The inverter AC output can go to a sub-panel that only has breakers for those items the inverter is expected to power - my preferred choice.
Another option is to go from the double pole main to the inverter AC input and then feed the whole panel from the inverter. This means that items like the water heater have to be off when using the inverter.
The battery switch they added on the DC input (positive cable I hope
) does make sense. There should be a fuse in the positive cable to the inverter as well - either a T class or ANL.
Below are 2 diagrams showing the wiring options described above.