Basically, the best way to think of them is that each pair of channels represents the left and right side of a single track. And the volume of each track is adjusted according to your location.
For example, with Kingfin in Super Mario Galaxy, only Channels 1 and 2 are playing, but if you get closer to Kingfin, Channels 3 and 4 (the drums) will start playing.
Meanwhile, in Beach Bowl Galaxy, Channels 1 and 2 will start playing, but when you're under water, Channels 1 and 2 fade out, and Channels 3 and 4 fade in.
It's a little more complicated with Koopa Cape in Mario Kart Wii.
On the main area, Channels 1, 2, 7 and 8 are playing.
When you get to the river, Channels 1 and 2 fade out, and 3 and 4 start playing, while 7 and 8 are still playing.
When you go down the pipe, Channels 3, 4, 7 and 8 fade out, and Channels 5 and 6 start playing.
Channels 1 and 2 are for the main part
Channels 3 and 4 are for the river section
Channels 5 and 6 are for the pipe section
Channels 7 and 8 are for that short "wah" bit playing the main melody.
In short, where O = on, and x = off:
Main section = OOXXXXOO
River section = XXOOXXOO
Pipe section = XXXXOOXX