He means with the battery removed. Basically, a bridge circuit will most likely cause the needle to rest in the center when no power is supplied. Where as in most other circuits, the needle will rest at the bottom (or top) indicating an underexposed (or overexposed) shot when no power is supplied. A bridge circuit basically has two paths and compares the two currents against each other, one being controlled by known resistors, the other being adjusted by the light sensor in the camera. So when the light hitting the sensor causes the second path to equal the first path, which is calibrated to be the value that is expected by a correctly exposed image, no current is flowing through the meter. Thus the needle galvanometer has no current running through it, so it comes to rest in the middle, just like it would if no voltage was being applied to the circuit. That's why bridge circuits don't depend on battery voltage, because if you increase the voltage on one path, it also increases in the other, but the ratio between the two paths doesn't change. It relies on the balance between the two paths instead of absolute values calibrated against an expected voltage.
Other types of metering circuits just take the current flowing through the by the light sensor in the camera and use that to directly power the needle's galvanometer. The galvanometer is calibrated to show a properly exposed image when the pre determined amount of current flows through it. So if you change the voltage of the battery, you'll change the amount of current flowing through the sensor, and will thus change the current flowing galvanometer at all light levels versus what is expected. That's why these types of circuits depend on using specific types of batteries.
Of course, there are easy ways to change the voltage of a battery, so those type of circuits can be adjusted for newer batteries. I suggested a resistor or zener diode, but I like E. von Hoegh's suggestion of using a tiny voltage regulator better. I've only ever used them for dropping a full volt or more in larger circuits, and didn't know you could get one for dropping such small voltages. But they're really the way to go for this type of job. Oddly enough, I can't find any zener diodes that would work for this type of application at such small voltages (I could have sworn I saw some once), so I was way off in my initial suggestion (though a resistor would still work, provided the new battery had a fairly steady discharge curve over it's lifespan).
I probably didn't explain all of that very well, and it's a bit more complicated than how I laid it all out. But that should give you the general idea of what's going on.