The items on the right column prefer to work with alternating current. That is provided by an inverter. The items at the right column may need to operate at night, possible when panels are not productive but their daylight production was stored in the batteries.
I think they should either use only the inverter or you should use the battery, both device store power that supplies to home devices. that's why I am confused,
No. Only the battery stores power (or more correctly energy). The inverter converts the panel DC voltage or the battery DC voltage to an AC voltage for AC loads (most mains-powered devices).