Hi,
sorry again for the late response. Like I said in the PM my response last week got lost somehow and I had a pretty busy week.
Yes, you will need an external battery for this circuit, as you said yourself between 3.3 and 5V. It is possible to run everything of the internal battery, but we would need to modify the circuit based on how the button is wired up inside the camera (either to ground or to V+). The circuit I gave you does not care about this and should work either way.
I will try to explain the circuit by describing what happens when a signal arrives.
1. When there is no motion detected the sensor outputs a LOW signal. The transistor will also receive a LOW input which means it is turned OFF and it does not conduct any current.
2. When motion is detected the sensor will output a HIGH signal. At first this signal is just passed through the capacitor to the input of the transistor, turning the transistor ON, which will allow current to flow through it.
3. Because of the resistor to ground there will flow a current through the capacitor. When a current flows through a capacitor, the voltage across that capacitor increases (it is charged, kind of like a mini battery). This makes the voltages on the input pin of the transistor slowly drop, which will turn it OFF, making it not conductive again.
4. When no more motion is detected the sensor outputs a LOW signal again. This will discharge the capacitor, which basically resets the circuit to it's starting position.
So the capacitor-resistor combination effectively shortens the pulse coming from the sensor. The length of this pulse can be tuned by changing the values for the capacitor and resistor, the higher the values, the longer the resulting pulse.
The transistor is in this case used as an electrically controlled 'button'. The resistor you use should work just fine.