ELI5: Presumably this was a result of the LHC's power upgrade to 13 TeV. If so, should they now be able to repeatedly get these results again and again?
Eh... despite what the other guy is saying, you can't really say this for sure. These bumps appear in a region that was accessible during Run 1 at 7 and 8 TeV.
See figure 3: http://arxiv.org/abs/1504.05511
If it is something real, and not just a statistical fluctuation (that's a big if), then it would have to be e.g. some decay product of a much heavier particle to account for why it wasn't seen during Run 1.
> If so, should they now be able to repeatedly get these results again and again?
Again, depends if it's a real effect or a fluctuation.