I know the theory you're talking about. I'm not an expert but I'll try and explain as best I can.
If you ask most people, the general consensus is that magic is caused by intent. When we cast a spell we are attempting to tip the balance in our favour or exert our will. Our intentions are obviously personal but it is hard to imagine anyone truly focusing intent without belief. Why would they bother? It's not so much that the spells are powered by belief but rather that the spells will fail without it. Aside from anything else, humans have this little peice of their brain that seems devoted to trying to prove our assumptions (whatever they are) to be correct. If someone without belief in magic tries to cast a spell, they will also subconciously be focusing the intent to prove that magic doesn't exist and thus the spell will fail.
As I said before, I'm not an expert. This is just what I understand from generally floating around the site and picking up ideas. I hope it helped.