The best I can think of is that plants are close to the ground and a "planta" is, from what I understand, a ground-level architectural plan, but that's just a shot in the dark. Aulete usually would help with origination but, in this case, it doesn't.
It looks as though it ultimately comes from the sole of your foot!
An architectural planta is a floor plan. My Dicionário da Academia das Ciências de Lisboa (DACL) says planta in all its meanings comes from the Latin word, well, planta, which meant a plant; a shot, twig, or sprout; and the sole of the foot. Now this doesn’t help much. The best dictionary for etymology I know of is the big Houaiss, but I don't have one yet.
I think the answer lies with the verb plantar. Plantar means of course to plant a tree or flower so that they can grow roots into the soil. But according to the DACL it also means to firmly set something onto or into the ground, such as pitching a tent, driving stakes into the ground, and laying foundation stones. These days we’d say montar a tent, cravar, espetar stakes, and assentar foundation stones, so plantar in this latter sense is probably old use, which is what we’re looking for here.
Now, to lay the foundations of a building you’d traditionally drive stakes into the ground (plantar estacas) and connect them with ropes or maybe poles to mark the boundaries of the foundation trenches. And then you’d lay stones (plantar pedras) in those trenches. So you’d make the actual planta or floor plan on a 1-to-1 scale on the ground. See this and this and this. I’m in speculative mode here, so you draw your own conclusion. But I’ll just add that forming the word planta from plantar in these senses would be very natural. Compare conta and contar (counting things), conto and contar (stories), fala and falar, and so on and on.
After all this I looked up the etymology of the French equivalent, and it looks as though the speculation above may be on the right track. My Hachette Pratique du Français says plan (planta, floor plan) comes from plant in the old sense of “assiette d’un édifice” (original quotation marks), which sounds like the plate (thing you eat from) of a building. And in assiette we have only assiette d’une route, plate of a road, which is the surface necessary for the construction of a road. The modern French plant also means the piece of ground planted with trees. Wiktionnaire says plan, non 2 (noun 2, the relevant one), was originally plant, our old friend the “plate of a buiding”, which was formed by removing er from the verb planter. Now planter has all the relevant meanings of the Portuguese plantar. So it looks as though plant, and most likely panta, originally meant the surface where you'd "plant" the building, and came later to mean a drawing of such place.
Both planter and plantar come from Latin plantare, which means to press into the ground with the sole of your foot (sola ou planta do pé in Portuguese). So Tchrist was right after all.