The short answer is, it doesn't. The Standard Model, AFAIK, does not rule out the existence of more neutrinos (or more generally, more "generations" of leptons and quarks than the three we know); there is no hard and fast limit in the math of the Standard Model on the number of generations. So I think the article's statement is, if taken literally, false. (Unfortunately, this kind of thing is common in pop science articles, even in magazines like Scientific American that used to be better at avoiding such things.)

I think a better way of stating our best current knowledge would be that we have only observed three generations, and if a fourth generation does exist, all of the particles in it would have to be very massive--massive enough to have not been detected in any of our current experiments up to and including the LHC. Most physicists appear to think it's unlikely that a fourth generation exists given those limits, but I'm not aware of any theoretical argument that definitely rules it out.