The light looks like it might be resting on top. Depending on the screen, it could be strong enough to hold (you just have to make sure the screen does not get too hot under it). Either that or hanging it from the ceiling could work.
The branches I heard are possible to hold in place with thumb-tacks, on the outside of the vivarium piercing into the wood from the outside. This is assuming that the wood fits perfectly inside!
::EDIT::
Just noticed the plants... the bottom one looks like it's just resting there. The others COULD be attached to a back wall that is not screened, but I am not sure!
I'm fairly certain that is Olimpia's cage. I would ask her if you want to know for sure but I would assume she is either using tacks, fishing line or just friction to hold them up.
Yep, that's my cage! The branches are help up by thumb-tacs, like someone else mentioned. One through each end is more than enough to hold up thin branches and the thicker ones have two. The screen does a great job of holding them up and they haven't ripped through, even after almost 2 years of use. I had someone hold the branch in place on the inside while I pushed them in from the outside and maybe finished setting them with a tap or two from a hammer.
And then plants all float in the enclosure. The top pothos is resting on some branches and the other two (bromeliads) are attached to the branches using fishing line (I use 20-50lb line, which is thicker). Nothing rests on the floor.
I recommend push-pins instead of thumb-tacs to anyone who wants to be able to remove the branches (for cleaning or to change the size as a chameleon grows, etc.) because the thumb tacs are hard to remove once they're hammered in. The push pin would be easier to remove without ruining the screen at all.