How does dust affect a computer?

It is generally known that dust makes computers slower, and that cleaning out the dust regularly can make your computer run better.

What I have not seen, is why this happens. From what I know dust doesn't stop electrons from getting from one side of the computer to the other. Does the dust only clog up the fans, making the computer hotter, thus making the computer slower? Or is there some other more mysterious force at work here?

Clogging up the fan is one side effect. It is also possible that dust can cause a short-circuit because it conducts electricity. Dust can also inhibit airflow, which prevents heat from being properly vented.

While a rare occurrence, dust can cause shorts. Mostly it will screw up your fans and coat heatsinks, which negatively affects heat dissipation.