To continue the epicycle analogy:
The problem in modern physics is that the epicycles work too well. There are some big obvious mysteries and an incredibly-successful epicycle-based theory. Essentially everyone expects that the epicycles are not the truly fundamental theory of things.
So, physicists keep comparing the predictions of the "epicycle-theory" with reality in new ways, ways where a deviation from the prediction would be an unambiguous signpost pointing a direction toward which one might be able to address a big mystery. As those tests become established, they are then performed with higher and higher precision.
The signal potentially seen with LHCb is, if it persists, important. In a nutshell, electrons are, in this particular way, predicted to act exactly like muons. No ifs, ands, or buts. What they see is that electrons and muons might be acting slightly differently. If so, some new physics (or a major error in our understanding of the epicycles -- I'm looking at you, loops) must be at work. If this result persists, it is guaranteed that we will learn something.
Too convoluted? Here is a different analogy:
You're pretty sure someone is embezzling from your company.
You look at your bank account, and you have ~95% less money than you thought you should. Really. Oh, no.
You check the summary budgets, they're balanced. So you do this for all the company's divisions. Yep. Still balanced. You get paranoid. You do this for decades -- the books balance, but at the end of the day? 5%.
Then an employee runs up to you and says, "Shmageggy! We're not sure yet, but it looks like one of our most-trusted partners, whom the firm has worked with since 1936, transferred thirty-seven cents to an account in the Cayman Islands!"
Until we find the embezzler, that's what every one of these precision physics tests that makes the news is all about.