It appears to be a result of the end of inflation. Basically, whatever it was that drove inflation had to have a whole lot of energy. When it decayed, it reheated our universe to a tremendous temperature.
Right, if the expansion rate would have been slower then, it would have. It would have progressed all the way to producing iron and that'd be the most common element.
But this takes time. First the protons and neutrons condense out of the quark-gluon plasma, so you have hydrogen right away. Then you start to make helium from the hydrogen. Then you start to make heavier and heavier elements.
As it turns out, the expansion rate was such that the universe cooled to where the nuclear fusion basically stopped by the time there was only around 25% helium sitting around, and long before more than trace amounts of anything else formed. Incidentally, this is even faster than it sounds, because much of the helium stemmed from the neutrons that were around early-on (when you have nothing but protons around, fusion takes a heck of a lot of energy, as you have to convert protons to neutrons, and because they repel one another, but when you have lots of neutrons sitting around it's much easier).