Things: people around here don't like people playing with mains if they don't know what they are doing. 120V mains, while lower then other country's mains voltages, is still lethal.
That said, from the schematic, the mains looks like it could be completely isolated from any breadboards. Please ensure you fuse mains.
You also need to tell us what tools you have at your disposal.
What I would suggest doing, is to break the circuit down into blocks on your breadboard, (and start off not having it powered from mains) first thing I would get working is the microphone input. If you have a scope, try see if you can trigger off the collector of Q1, to ensure that side of the circuit is working, then add in additional parts to the circuit.
I would second the suggestion----
This isn't really a high frequency circuit,so you should be able to check the Bistable & the other stuff with any old DMM.
The only place you may have problems is with the mic circuit,because the pulse caused by the "clap" is only of short duration.
The circuit operation description on the website is pretty lousy,& I can maybe do better,so here goes!
Assume initially,that
Q3 of the Bistable multivibrator (flip-flop) Q2,Q3 is turned "on"hard (saturated),so its collector voltage is very low,approaching zero volts
At the same time
Q2 is "cut off", so its collector voltage is very high,approximately at the positive supply voltage.,
Q3 collector is connected to
Q4 base via D4 & R13,&
Q4 is "cut off"so the relay is not operated.
If the Bistable M/V changes state,
Q2 is now "saturated",&
Q3 is cutoff.
Q3's collector goes high.turning
Q4 "on",& operating the relay.
So far,so good!
How do we make the Bistable M/V change state?
The mic "hears" the "clap" generating an ac signal at the base of
Q1.
The amplified signal from
Q1 is then fed to Q2,Q3,bases via C3,D2,& C4 ,D3.
The diodes only pass the negative half cycle of the signal,presenting the bases with negative pulses
which momentarily overcome the existing positive bias on the "saturated" transistor,(in this case,
Q3) & turn it "off".
Q3'scollector goes "high",biasing
Q2 "on".
PS: I wouldn't panic much ,as far as the transistor types for the Bistable & relay driver are concerned,but Q1 is an amplifier.
If it has insufficient voltage gain,its output may not be sufficient to trigger the Bistable into changing state.
Another point is to get a mic which has a fairly high output voltage.
Apparently they used an Electret mic,as they have an FET preamp built in,hence.R1.