I think it has to do with the equivalence between mass and energy, at the fundamental, quantum level.
Correct. To answer the original question, think about mass-energy equivalence (E=mc2). What the LHC did was smash protons together with enough energy to cause a Higgs boson to be created. The mass of the protons aren't really as important as the energy involved in the smashing. Think about it - what has a bigger impact, a semi-truck rear ending another semi-truck at 5mph or a Mazda Miata rear ending another Mazda Miata at 100mph? The point is that they needed to create enough energy to cause a Higgs to be created, which it turns out just so happens to take 125GeV.
See, they increased the energy on two protons beyond 125 GeV (where 125 GeV is the energy-equivalent of 125 protons, give or take).
I'm a bit confused by this part. A proton is made of two up quarks and a down quark. Using Wikipedia's top-end estimates of the mass of those three, that means a proton should have a mass of at most 11.9MeV which would make 125 protons weigh 1.4875GeV... nowhere near 125GeV. I'm not a physicist so I trust Mr. Organtini, but I can't figure out where this figure is coming from.