That was confusing to me as well when I have received a similar advise.
Voltage is not an absolute value, it is always referred to something else. Like if I say
this table is at 1m: it will mean nothing unless I provide a reference, like
this table is at 1m from the floor or
1m above sea level (the world wide considered floor, or 0m).
Now let's say we put a chair, which feet are 50cm height, between the table and the floor: I can say
this table is 50cm above the chair and the floor is 50cm below the chair! We got a negative measure!
Let's apply this to the voltage thing: if you have a 12V battery input we, by convention, associate 12V to the anode (+) and 0V to the cathode (-). I bet you know voltage dividers, so you know you can easily obtain a middle point by connecting two equivalent resistors in series between your battery cathode and anode to achieve three voltage levels: 12V, 6V and 0V. Now shift your convention point from the bottom to the middle point and what you get is +6V, 0V and -6V. Obviously you will have to use the 0V point as GND in the rest of your circuit for the battery cathode to be considered -6V
If the last statement isn't clear enough, that means your board GND (and anything else you want to be at GND level) will have to be connected to 0V, including the opamp reference input. The only connection you need to have at -6V level is going to be your opamp negative supply.
Hope this clears it up
P.S.
There is an equivalent of
sea level for voltage and it is called
Earth so it has a sense to say
this circuit has earthed ground, sometimes abbreviated to
grounded and its meaning is the circuit's 0V corresponds to our planet electrical potential. It's a convention, as much as the sea level thing, as we know we can go below sea level and get negative values
