When you move deeper into the bowls of Edanna, you hit an open area with a bridge. At the dead end on the other side of the bridge, turn on the orangish plant by clicking on it. It'll cause the green plant to unroll for you. Now go over to the swinging plant. You can't swing yet, though.

If you were stuck getting to this point, go try the rest on your own, it's sort of fun :)

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OK, if you're still stuck doing the swing part ... look right. There's a tent there. Turn the crank to raise it, and then tap the plant to release a nut. Then click on the nut to move it. Head over past the crank, down to the other end of the now-rolled-up green plant. Pull the handle there to drop the tent, it'll scare the creature, it'll run across the bridge, taking it down. Now you can swing on your plant across to the next area.

The Myst 3: Exile Walkthrough