"BZ" is the buzzer (loudspeaker), the easiest way to make it quieter is to place some tape over the hole in it. The tape doesn't have to be very thick, the body and hole size determine how loud it is (they form a resonant chamber) so anything that detunes it will cause a dramatic drop in volume.
Unless it uses a capacitive dropper power supply, there must be some components we cant see. If it IS a capacitive dropper, I would guess R2 and C1 are the dropping components and the small 8-pin IC is a switching voltage regulator. L1 looks like it could be the storage inductor. Can you tell us what the number is on the IC (U1), I think it says Viper12A but the picture isn't clear enough to be sure.
A word of caution, if I'm right about the power supply, it does not provide any isolation from the incoming AC wires and the product is only safe to the user because the live parts are out of reach. With the case open, even wires that appear to be low voltage could have more than 300V on them with respect to ground so they are potentially extremely dangerous.
Brian.