Bittersweet seeds develop inside these berries. When the mature berries are eaten by birds the seed portion of the berry passes through the bird's digestive system undamaged and is later (for lack of a better term) pooped out, fully viable and ready to develop wherever it happens to fall. Or if not eaten the fruit simply drops from the vine to the ground. Either way it will appear the following year as a new seedling.
This is a major reason to cut oriental bittersweet vines ahead of the maturation of the berries. Once cut and deprived of nutrients the above-ground portion of the cut vine will die. Immature berries will also die before becoming viable, although nutrients already in the vine can continue the maturation process for a short period after vines have been cut. The preferred window of time for cutting vines will be closing as we get into mid-summer. If you can, get out there and do your cutting ahead of the berries maturing.
Cutting vines with mature fruit on them is still worthwhile but the vines will drop viable seeds unless you pull the vines down with the berries intact and dispose of the debris in a manner that prevents the berries from developing into next year's crop of seedlings.
Always carefully inspect trees before cutting any vines that have grown up into a tree. Look for power lines, unsafe branches, weaknesses in the tree itself, and vines that may have crossed into other nearby trees. When in doubt, hire a professional. Pulling the vines too hard can damage tree limbs or worse, bring a hidden branch or entire tree down on your head. Be careful and if some gentle tugging doesn't work, leave it. The vine will die and eventually fall out of the tree on its own. All the more reason to deal with oriental bittersweet well before it gets up into the trees.