When you use one battery it will be at a certain voltage. The device will use this voltage, the maximum power available from the battery will depend on that battery/what it is made from etc. A battery grip will offer the power of the two batteries to the camera in SERIES which means that the camera will get the SAME voltage. It is also possible for simple electronics in a grip to switch between batteries. If your 6 AA batteries are placed into the grip the maximum voltage would be (alkaline batteries) 1.7v x 6 = 10.2v or (NiMH batteries) 1.4v x 6 = 8.4v the camera's circuitry will be designed to work with a range of voltages converting them to the voltage required internally.
Within a range a device will draw what power it requires so if you have 1 x aa battery and connect it in a simple circuit to a bulb it will light up and glow away merrily for 10 hours (for example). If you connect 2 of these batteries together in series (IE pos to pos and neg to neg) then you still get 1.4-1.7v and the battery will still power the bulb at the same brightness but it will last 20 hours.
If you connect the two batteries in parallel (IE neg of one battery to pos of the other then the two ends not touching each other into the circuit) you ADD the voltages so you get 2.8v to 3.4v this will most likely blow the bulb up unless the bulb can handle it in which case it will glow much brighter but only last for the same 10 hours.
(note this is all rather simplified but it does sort of explain it)