You don't have to understand the theory behind the QuadraFuzz to build it or use it but the circuit demonstrates a lot of basic principles of musical electronics.
IC1B is a preamp which accepts either a high level input (via J1) or a low level input (via J2). ATTACK control R52 sets IC1B's gain from 2 to 200. At higher ATTACK settings, this stage introduces some broadband distortion of it's own that is particularly useful for lead sounds.
This preamped signal then splits (via R29 through R32) into four bandpass filters. The filter built around IC2B (LO) is tuned the lowest, the one built around IC2C (MID 1) is tuned somewhat higher, and the filter built around IC3B (MID 2) is tuned approximately one octave higher than MID 1. The HI filter, IC3C, is tuned one octave higher than MID 2 but also offers a switch-selectable bandpass or pseudo-highpass response. The other filters include switches (S2 - S4) which allow for increasing their resonance (sharpness).
The filter outputs then go to their own individual fuzzes, built around IC2A, IC2D, IC3A, and IC3D. These fuzz stages use LEDs rather than the usual diodes for the clipping (distortion-introducing) elements; I did so because LEDs clip at a higher voltage, which allows for more output from each fuzz stage. LEDs also seem to clip in a somewhat gentler fashion than regular diodes, thus giving what is probably best described as a more "rounded" sound.
Each filter output can be individually tapped (note that these outputs are in phase with the master output, so you won't run into cancellation problems if you use an external mixer to combine the individual outputs and master output together). Each filter output also feeds its own level control, so you can regulate the balance of the various frequency bands. The level controls go to a mixer (IC1A) which combines the signals together prior to feeding a tunable active filter (IC1D). This two-pole lowpass filter provides additional control over the final sound, and seems to work well with guitar. Also note that loop send jack J10 and loop receive jack J11 bypass the filter, thus letting you patch other types of filters or signal processors in place of the Quadrafuzz filter if desired.
The filter output proceeds to FUZZ LEVEL control R18, which feeds the output mixer/electronic footswitch stage (IC1C and IC4A-D). The four schematic symbols that look like a cross between a pushbutton switch and a FET represent the individual FET switches inside a CMOS 4016 IC. These are arranged so that with the fuzz bypassed, FET switch IC4C is open (thus setting the preamp to minimum gain), and IC4A is closed (which allows the preamp output to feed into IC1C). When bypassed, IC4B is also open. With the fuzz effect active, IC4C closes to allow for adjustment of IC1B's gain, IC4A opens to cut off the straight signal, and IC4B closes to let through the fuzzed signal. D9, the status LED, is an integral part of the switching circuit. If for some reason you don't want to use an LED, then replace it with a 10K resistor.
IC5 and IC6 are regulators that provide a stable power source to the Quadrafuzz. They accept a bipolar supply voltage in the range of ±5V to ±20V and regulate it down to ±5V DC. Diodes D10 and D11 provide protection if the regulator output exceeds the regulator input voltage (a condition that could otherwise damage the chip).