We will go through it one section at a time, please refer to the schematic.
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.'
Class B 'pushpull' amplfier, which uses two seperate transistors to amplify the sound.