Okay, the circuit is a little odd, so we'll go through it bit by bit. You very quickly learn to recognise "blocks" of circuits, so when you see something you can kind of guess what it's supposed to do. It's a bit like, even if you don't entirely understand a word you haven't seen before, you can figure it out from the context.
I don't know how much you know about electronics, but there are masses of sites out there that will tell you all about it. So I'll assume you know what resistors and capacitors are, explain opamps briefly, and crack on.
The circuit uses a bunch of opamps (Operational Amplifiers) which are chips that can be wired up to do various clever things (not just amplifying, as we shall see!). Look at the top left corner, and you'll see IC3 used as an amplifier (with a twist which we'll come to) and a buffer - the buffer has no gain but "isolates" the input from the load on the output. In the datasheet for the opamp it describes pins 1, 2, and 3 as being opamp A, and 5, 6, and 7 as opamp B - dual opamp, two in one package.
Okay, IC3, then. You tap the piezo beeper - <BONK> - and you get a big wobbly pulse. The sensitivity pot controls how much of this is fed to the amplifier formed by one half of IC3, through a simple highpass filter formed by the capacitor and 22k resistor. The signal goes in the non-inverting input, where it's amplified by 1000 (that would be 4M7/4k7 = 1000). The clever twist is the diode, which ensures the output pulse only ever is positive, and the output of the opamp doesn't discharge the envelope cap. C8 charges up from the pulse, and discharges through R17 and the decay pot - R17 ensures that there's a minimum decay time and we don't just short the cap to ground. NOTE: It's a very good idea to use a resistor in series with a pot if you're adding a pot to ground to fuzz up a circuit that you're bending. It might save delicate parts from getting fried. That buffer we mentioned earlier consists of the other half of IC3, with the inverting input directly connected back to the output, and the signal fed to the non-inverting input. The output follows the input exactly, but if you load the output with something it won't affect the current at the input. Think of it as power steering for C18 ;-)
I'm not totally sure what the Attack circuit does, so I'll worry about that tomorrow. I *think* it adds a bit of a "click" to the output, but I don't know. Anyway, IC2 forms a VCA, with the control voltage from the decay control being fed in through R19. It's a special kind of opamp called a "transconductance opamp" - what this means isn't very important but basically you can control how much gain it has with an external source going into pin 1.
The funny bowtie and semicircle symbols indicate solder pads that can be cut or bridged to change the sound. You could break these out to switches. Anyway, we'll move into the good bit - the oscillator.
The oscillator is formed from IC4A and IC5B in the middle of the circuit. It works like this. C15 charges up through R24, and the voltage on pin 1 increases as it does. IC5B is wired as a comparator, and when the output of IC4A reaches a certain point the output of IC5B turns on, which turns on the transistor, which starts to discharge C15 again through R29. Eventually the voltage drops far enough that the transistor turns off again and C15 starts to charge again, and we're off round again.
The rate at which C15 charges is determined by the voltage on the output of IC4B. It has various inputs from the LFO, envelope and pitch control. The voltage on pin 7 of IC4 should vary when you turn the pitch knob, trigger the drum, or increase the LFO mod. If it doesn't, it might be that one of the pitch or mod controls is miswired. The build instructions are out there on the 'net, along with the circuit diagram. It's probably worth checking through the build instructions to ensure that everything is wired up as it should be, before you start.