How does a Logitech scroll wheel detect movement?

There's a diode (probably infrared) and a receptor, one on either sides of the wheel, soldered directly into the green PCB. Wheel is SPOKED, and by turning it you interrupt light travelling from the diode to the sensor. No magic there.

Oh, and dust sitting in there can permanently block sensor from any light coming from the diode, so there's no way for it to register interruptions, that's why the wheel appear to not work.

To answer your edit and expanding AAronD's excellent comment, the direction is sensed by a microcontroller by the sequence of the codes when the wheel is turned.

For example, the two diodes, when turning the wheel, let's say, counter clockwise will translate to the following sequence:

    A B
    0 0 
    0 1 
    1 1 
    1 0 

For the clockwise direction, the inputs will be shifted in the oposite direction.

An excellent graphic illustration is also available here: http://www.creative-robotics.com/quadrature-intro

I did some research and found out that there are many types of scroll wheel types. It is not just Logitech , but almost all mouse manufacturers use this type of scroll wheel.The one here is optical scroll wheel. This uses a light shining onto a photodiode through slits in a metal or glass disc. Reflective versions also exist. This is one of the most common technologies. Optical encoders are very sensitive to dust.

The optical encoder's disc is made of glass or plastic with transparent and opaque areas. A light source and photo detector array reads the optical pattern that results from the disc's position at any one time. The Gray code is often used. This code can be read by a controlling device, such as a microprocessor or micro-controller to determine the angle of the shaft.

The absolute analog type produces a unique dual analog code that can be translated into an absolute angle of the shaft

I faced this problem last week and after opening up the unit and toying around a bit I was able to fully repair it.

The main problem is the slimy liquid that reaps out of the ageing rubber, the scroll wheel is wrapped around with this rubber grip. Scroll wheel with spikes If you see carefully the wheel has spikes that allow the optical sensor unit to kind of see through it and it is due to these variations while rolling the wheel the sensor is able to detect if you are performing a scroll or not. Now the slimy liquid will reap into these spikes and block the sensor's view entirely which will make the firmware unable to detect the scroll action.

The solution is simple just remove the scroll whell and the rubber grip, and clean it with a soap solution, dry it and make sure you are able to see each spike clearly when put against the light source.

And you will find that it will be as good as brand new.