To eat a muscadine, place the grape with the stem scar facing upward in your mouth and squeeze or bite the grape. The pulp and juice will burst through the skin into your mouth. You have several options for what do do next, as there are several schools of thought, and this is often much argued over!
You can take the skins out from your mouth right away, OR you can chew the skins lightly first to get more juice out, OR you can chew up the skins and swallow them. Some varieties have skins that are thinner and more flavorful than others.
You can swallow the pulp and seeds together OR you can work the seeds out of the pulp with your tongue, and spit them out before you swallow the pulp.
For the greatest nutritional benefit, eat the skins and seeds as well as the pulp and juice. New varieties are being developed that are seedless and have more edible skins. So far, these are mostly available for home garden planting, but watch for them in the market, too.