Someone identified what looks like an IC here:
Identify unknown SMD component code A6530 72130
Looks like a type of fuse.
Yeah, I know this a really late reply, but seeing as how no one's really answered it correctly, I'll chime in now (heh, pun not intended).
So, looking at the datasheet for the component in the August doorbell power board, it is certainly a fuse. How it works is like this:
Wired in parallel with the chime, it shunts part of the current through the power board. The rest of the power goes through the chime. The fuse is about 10ohms, so it's like putting in the same value bypass resistor.
If the electromagnet in the chime is about the same resistance (we'll ignore impedance at our own risk since this is an AC circuit), half the power goes through the chime and the other half goes through the fuse. Hopefully that lower amount of current is not enough to move the hammer enough to hit the... chime. Okay, let's call it the "dinger." If it is, just add another power board to drop the power through the chime to a third. (I'll be using the words "power" and "current" interchangeably, though that's not completely correct)
Now, when the button is pressed, there's a circuit in the camera that just shorts out the power through the camera unit. This causes a surge in current that blows the fuse, sending the full amount of power through the chime's electromagnet, causing the hammer to strike the dinger and giving the "ding" sound.
I'm not sure if the doorbell's short circuit is interrupted by the button (presumably not, or you could disable the camera by holding down the button until its backup capacitor fully discharges) or some other internal mechanism.
Either way, when the camera's circuit cuts the short circuit, the fuse resets (okay, I guess it's a self resetting circuit breaker) and current flows as it was, lower through the chime, allowing the hammer to release and probably overshoot backwards hitting the secondary dinger, the "donger," giving the "dong" sound. Too much current through the electromagnet and the hammer wouldn't "dong."
So the power board is a slight improvement on just a regular load resistor in that it allows more current through the chime when the button is pressed so you get a more forceful "ding" sound. The power board and load resistor both keep the chime from buzzing and lightly chiming when the button isn't pressed.
Yeah, I know, the instructions don't say to put the load resistor in parallel with the chime (though I think others have found that to work), but rather in series with the camera and power supply when there is no chime. This is to prevent burning out the home wiring when the camera's power lines are shorted together when the button is pressed. Otherwise you'd have something like 10 to 40 amps going through wires not designed for that much current.
Okay, I've assumed people here know how a doorbell chime works. If you don't, there are plenty of explanations available with a simple query.