Updated December 29, 2015:
Matt Strassler engages in a similar exercise in a somewhat more accessible manner. He starts by making clear that it couldn't be a Higgs boson, because if it produced that many diphoton events (which is can't produce directly since photons only couple to charged particles and this is electrically neutral) it would produce far too many top-antitop quark or tau-antitau pairs to escape notice at the LHC until now.
So, there needs to be some new undiscovered intermediary particle.
If this bump is a real particle, there must be at least one other particle that makes such a two-photon bump possible. That new particle (maybe heavy, or maybe lightweight) hasn’t yet been found by the ATLAS or CMS (or LHCb!) experiments, but is almost certainly accessible to them. In fact, signs of an additional particle or particles may already be obvious (or almost obvious) in their existing data. It might just be a question of looking in the right place and of asking the right question. . . .Of course, the lack of a theoretically well motivated particle that could produce the 750 GeV diphoton excess is one reason that many people think that this might simply be a false alarm and statistical fluke.
There are a lot of possibilities. It could be a simple elementary particle like the electron, interacting with known particles indirectly, as a result of direct interactions with some heavy, as yet unknown particles. It could be a composite object like a proton, made from objects bound together by a new force that we are about to discover. It might be that the photons to which it seems to decay aren’t photons after all, but are new particles that mimic photons in some way (an idea that goes back at least to 2000.) I certainly don’t know what it is, and neither does anyone else yet. But each of the many different ideas comes with predictions — predictions which in many cases lead to LHC signals that can be looked for NOW!