The nature of the experiment has been discussed here, but my main question is this: is there any theory that has predicted the results of this experiment or are we completely clueless about what's happening? In other words, have we come up with a new hypothetical interaction that could explain the results?
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In the standard model the $α=(g-2)/2$ of an elementary particle should be calculable , the calculations as accurate as the higher orders are computed.
For the electron the calculations coincide with the experimental value to great accuracy
The muon $α=(g-2)/2$ has different diagrams dominant so the theoretical value will be different, but it was seen ,first at the Brookhaven data, that the theory differed , not statistically significantly, with the data and so the Fermi lab experiment was designed at great effort experimentally and expense, because the difference would mean that a new interaction should add to the effect, not in the standard model that the calculations use.
As the comments say there are many extensions of the standard model that can try to fit the new results and thus choose the extension or the new theory. Hit the link given by G.Smith to get a large number of papers using different theories.
anna v
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A Nature paper was published on the same day, which seems to have attracted a lot less press. This presents a recalculation of the muon $g-2$ value, using standard model physics and their value is consistent with the new experimental value (Borsanyi et al. 2021). So there's one theoretical explanation of the result!
ProfRob
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Yes there is a theory that explains the results ... the Standard Model. In other words, the claim is that the Standard Model already is consistent with the experimental data, and the original "prediction" was calculated wrong.
Check out the paper published in Nature together with the muon g-2 results, or the writeup at popular level.
Allure
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This answer was already suggested above. This is another source that has some really nice embedded links on the authors and the university that hosted the study. Very nice reading some might enjoy so I thought it would be nice to share. Assuming I am allowed to put in a link. Not sure it is allowed by stack exchange.