There are no epiphytic vines in the US. Most likely, if it's sturdy enough that you can swing from it, it's the vine of a wild grape. (The vine in the video is classic wild grape.)
They grow like most other flowering plants: they start innocently enough with a seed that takes root in the ground. They are initially tender and small stemmed.
From there, they grow along the ground, sending out tendrils. If they find something growing vertically, they cling to it (a shrub, tree trunk, or a low hanging branch) and grow upwards towards the light. If they have enough light (they are shade intolerant), they continue to grow, branch, and elongate. because they can become quite heavy, they break canopy branches of their supporting tree and fall in long loops toward the ground.
If the loop breaks for some reason, you have one swinging vine. But if you diligently follow it back, you'll find either the continuation of the vine near the canopy (the part after the break) or the trunk growing from the ground alongside of a nearby tree.
Though cool-looking, they are very destructive, bringing down not only branches but often entire trees (plural. They intertwine different tree branches, and if they bring down one tree, others fall in a domino-effect.)
Imagine that SOB hanging onto your upper arms for support (not the guy, the grapevine).
Smaller vines (which can also be destructive) that climb up the sides of trees are Oriental Bittersweet, Poison Ivy, Virginia Creeper, etc. (Probably Kudzu as well, but we don't have that here yet.) But these aren't likely to become large or sturdy enough to swing from if they come loose from the tree.
This is a photo of two common offenders: a wild grape (dark shreddy bark, about 2.5 inches in diameter) being slowly strangled by an oriental bittersweet (lighter bark), both climbing this large oak. If you look carefully, the base (roots) of both these vines are behind the tree (about five feet), one on either side. (You might have to take my word for it...)