Simply put your garlic clove, with paper and all, in the cup of the garlic press. (This might be where I was going wrong before, since I always peeled garlic before trying to press it.)
And squeeze. There's some good leverage action here, so squeezing is easy.
The garlic is extruded from the perforated end.
Scrape off the extruded garlic, then open the press and remove the paper. If you don't press too hard when you're squeezing, its easy to peel off.
And you easily have a pile of minced garlic, ready for whatever you're cooking.
You can do this directly over your pot or pan, where the juices that are pressed out can fall right into what you are adding the garlic to. I think my fingers are especially thankful - I'm clumsy and have to be very careful not to nick myself when chopping, especially something fine like minced garlic. Love it!