Or the digital way: on the Commodore 64 disk drive (the 1541), you'd flip a bit of a certain memory-mapped location to switch the LED on or off. Doing this a variable amount of times per second would result in a smoothly fading led. Not sure if anyone ever made a sinusoidal fade, but there were demo's with the LED following the beat of the music.
Pulse width modulation (PWM) is a pretty common technique. This article seems to be more about measuring the actual waveform Apple uses. It even says that the PWM used in the original introduced artifacts into the measured waveform.
I saw someone do exactly this with a 1541 at a party in the summer of 1990: up, down, up, down. It didn't quite do Apple's special rhythm, naturally, but it was quite a shock the first time I saw it.
LEDs also don't change brightness with voltage, they do with amperage and usually have an optimal amperage for the LED's design. Brightness can be controlled many ways, the most common way is to use Pulse Width Modulation. It's also a common power saving technique.
