Even though I aced the game both as a kid and later on as an adult, I was never able to fully understand the "rail grind" mechanism.

How do I do it? For example, the green rails in the tutorial?

You need to press the Luge button and keep it pressed. Then also press the Stall button and keep it pressed (= "stalled Luge"), meaning both buttons are then pressed.
This will more or less automatically attach you to the rail.

Or whatever else you assign to your keyboard/controller.

I failed many times until I figure out that you have to make sure not to press accelerate button.

Sadly the tutorial does not explain it properly. Check out the manual.