It is usually easier to leave the inverter switch switched to on all of the time, and then just switch the power that you supply to the inverter. Use a relay rated at the current you require.
As for your question, Earth is ground, supply is +12, load is the wire to inverter, thus most likely connection is to ground the earth, red from the inverter to the supply, white connected to the load (you should check with a voltmeter/test light the red and white wires. One of them will have a constant 12V on it all the time, the other won't. The one with 12V goes to supply, I'm guessing that it is the red wire.). You apparently have purchased a lighted switch, that is why there are those three connections.