The electrons and the muons are very similar. We can measure the magnetic moment and make some calculations and calculate a number g. If they were perfectly ideal particles, then g must be exactly 2, so it's interesting to measure g-2.
The real particles have a lot of virtual particles that appear around them and are impossible to detect directly. It's like a cloud of more electrons, positrons, photons, and other particles.
They are impossible to detect directly, but they affect slightly the result of the experiments, so when you go to a lab and measure g, you don't get exactly 2.
We have a very good model for all the virtual particles that appear around them, i.e. the electrons, positrons, photons, and other particles. It's call the "Standard Model". (But I don't like the name.)
We can use the "Standard Model" to calculate the correction of g of an electron, and the theoretical calculation agree with the experiments up to the current precision level.
We[1] can use the "Standard Model" to calculate the correction of g of a muon, and the theoretical calculation does not agree with the experiments!!!
The disagreement is very small, and there is still a small chance that the disagreement is a fluke, but people is optimistic and think that it they continue measuring they can be confident enough that it is not a fluke.
[1] Actually not me, this is not my research area, but I know a few persons that can.
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Back to your question:
> *Why is this important?
If the theoretical calculation and the experimental value disagree, it means that the "Standard Model" is wrong. Physicist would be very happy to prove that it is wrong, because they can study variants of this experiment and try to improve the model. (And be famous, and get a Nobel prize.)
Physicist are very worried because they are afraid that the "Standard Model" is so good that to prove it is wrong they need to build a device that is as big as the Solar system. (And they can't be famous, and the Nobel prize will go that work in other areas.)
If this result is "confirmed", the idea is to add a new particle to the "Standard Model" and get the "Standard Model II". (IIRC it already has a few corrections, so we will call the new version the "Standard Model".)
It's difficult because the new particle must change the predictions for this experiment, but not change too much the predictions for other experiments. It may take a few years or decades to find the new theoretical particle that match the experiments.
If you are pessimistic, the new particle will be useful only to explain a small correction that is only relevant in very accurate experiments in the lab, or inside a big star, or other unusual events.
If you are optimistic, in 100 year every moron on Earth will have in the pocket a device that will use this new particle for something amazing.
Or perhaps something in between. Nobody has any clue about this.