There are a few different ways. You can use fishing line for lighter things like vines. You can use thumb tacs and push pins to put in to the ends of beams/rods through the mesh to secure it without damaging the screen. The most popular way seems to be to make a frame within the cage out of bamboo or dowels and then secure things to the frame using wire or zip ties. This supports the weight of cups and plants and heavy things without risking damage to the screen. Check out photos of other enclosures here and you can spot these. Everyone seems to have their own way of doing it.
