The Drawdio kit is pretty simple compared to most electronic projects.
We will go through it one section at a time, please refer to the schematic.
The first part to understand is the power supply, which keeps everything running. The power source is a single AAA battery which is held in a plastic container that is soldered to the PCB. There is also a switch S1 which can connect or disconnect the battery from the rest of the circuit. Finally there is a large electrolytic capacitor C4, which is used as a 'bypass capacitor'. This means that it goes across the AAA battery and smooths out any ripples that are caused from power surges (like when the drawdio makes noise!)
The heart of the kit is a TLC551 chip, which is a low voltage version of the famous LM555 timer chip. This integrated circuit is specifically designed for creating timers and oscillators. The original LM555 needs at least 5V to run (which is more than 3 1.5 batteries!) so we are using the 551 which can run as low as 1V. It's a little more expensive but makes the whole thing fit on a pencil.
One of the more popular ways that '555s are used is as a 'astable multivibrator' which is anothe way of saying an oscillator. The frequency of the oscillation is set by 2 resistors and a capacitor. The chip slowly feeds current into the capacitor until it is full and then, likewise, slowly drains it out. The resistors set how fast to fill and drain the capacitor and the size of the capacitor indicates how long it takes before it fills.
This system is pretty much identical to Japanese water fountains, as this video shows:
f = 1.44 /(C3 * (RA + 2 * RB))
C3 = 680 pF = 0.00000000068 F
RA = 10000 ohms
RB = 300000 ohm to 1000000 ohm (1 Mohm)
In this case, we use two resistors RA and RB but also have an 'open connection': two tabs at the end of the PCB. If the two tabs are shorted with a piece of wire, then the frequency of the oscillation is 3500 Hz, if instead there is a resistor of 1 Megaohms between the two tabs, the frequency of oscillation is about 1000 Hz. You can use a calculator to do the math yourself using the formula above.
Instead of a 'everyday' resistor between the tabs, however, we use something else - the conductivity of the human body and graphite! The human body has a resistance similar to a 200,000 ohm resistor. Graphite has resistance of about 1 ohm per inch when it's in the form of a pencil lead. When it's spread out on a piece of paper, the resistance goes up a lot, up to a megaohm for a few inches of 'drawn resistor.'