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I'll do my best!
Just to set the scene briefly: all known material stuff in the Universe is made of chemical elements, all elements are made of atoms, all atoms are made of protons, neutrons and electrons, and all protons and neutrons are made of quarks (specifically, one proton = two up quarks plus one down, and one neutron = one up plus two down). Particle Physics tries to explain how quarks and leptons behave and interact with each other.
For the purposes of understanding this result, there are six particles we need to know about:
1. The electron, easily the most familiar to most people, as this is what carries electric current.
2. The muon: possibly the first truly weird particle to be discovered. It's essentially just an electron but about 200 times heavier.
3. The strange quark: another weirdo, first seen in so-called "strange" particles, because at the time scientists didn't understand their behaviour.
4. The bottom or "beauty" quark, which is again effectively just a heavier version of the strange quark.
5. The K meson, which contains exactly one strange quark.
6. The B meson, which contains exactly one bottom quark.
That's a lot of scene-setting, but hopefully it all makes sense.
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Next, there's a basic rule in particle physics that heavy stuff decays into lighter stuff. In particular, the B meson is the heaviest thing I have mentioned so far, and it's allowed to decay, among other things, into:
1. a K meson plus two electrons, or
2. a K meson plus two muons
These decays don't happen that often, because they involve a bottom quark turning into a strange quark, a process that the Standard Model says can't happen easily. On the other hand, processes that are expected to not happen very often are the most interesting, because if anything new happens it is likely to make the decay more common! That's why physicists have been looking at this process in particular, it's been on the radar for some time.
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The final thing to know is that, other than the muon being 200 times heavier than an electron, there is supposed to be no difference between an electron and muon: the Standard Model treats them as the same thing in all other respects. Therefore, the two decays above *should* behave in more or less the same way, and in particular there should be the same number of each. To make things easier, physicists therefore defined the ratio:
R_K = (number of times B decays to K plus two muons) divided by (number of times B decays to K plus two electrons).
The prediction is that R_K = 1. This experiment has shown that R_K = 0.85, which is (a) not 1, and (b) far enough away that it seems to be a genuine result, rather than just experimental error.
I haven't proofread this, but I hope it's clear and that it helps. If it isn't clear, hopefully it is at least enough to guide further questions.