They don't expect it to show up in every energy band - the whole point is that they expect to see a lot of Higgs boson formation at a specific energy level, and anything higher or lower drops to a background level. The sigma value is basically a measure of how many standard deviations off from the background event count the event count for this particular energy level is.

I'm sorry but I don't see how this addresses my point? If they are checking 20 bands for results with p < 0.05, the likelihood is that one will show up, just from fluctuations from background.

Yes, but there's a prior prediction from theory that the Higgs will be within this energy band.

This specific one? What about all the other predictions over the years?