Do you have holes drilled in the bottom? You should -- these will allow excess liquid to drain out. If you have it elevated off the ground (e.g. by a couple of bricks underneath it), then you may get a little extra air via these holes.
I'm skeptical of the idea that small holes drilled into the sides of a large (15-30 gallon) container filled with compost will introduce much air into the center of the compost; thus I don't think it matters much if you have your container elevated.
Composting does depend on the microbes having adequate oxygen, but it will be much more effective to aerate the compost -- just shake the container as recommended in the article you linked, or stir it with a garden fork -- after it starts to cool down.
The only other benefit I could see to having it elevated off the ground is if you want to catch the leachate and use it as a liquid fertilizer. To do this, put down a catch pan, put a couple of bricks on the catch pan, then put your bin on those bricks. Whatever drips out of your bin will collect in the pan, and you can pour it off to use it on your plants.