Yesterday, a team of physicists from France announced a breakthrough in nailing down a "magic number" by adding three decimals to the the fine-structure constant (news article; technical paper)
$$\alpha^{-1}\approx 137.035\,999\,206(11)$$
To the layman's eyes, 3 more decimals does not seem so spectacular. Why is this such a big deal when it is about the fine-structure constant?
The fine structure constant tells us the strength of the electromagnetic interaction.
There are some misleading statements in the news story. The big one is how to read the result,
\begin{align} \alpha_\text{new}^{-1} & = 137.035\,999\,206(11) \\ &= 137.035\,999\,206\pm0.000\,000\,011 \end{align}
The digits in parentheses give the uncertainty in the final digits; you can see that the traditional $\pm$ notation is both harder to write and harder to read for such a high-precision measurement. The new high-precision experiment is better than the average of all measurements as of 2018, which was
$$\alpha_\text{2018}^{-1} = 137.035\,999\,084(21)$$
You can see that the new uncertainty is smaller than the old uncertainty by a factor of about two. But even more interesting is that the two values do not agree: the new result $\cdots206\pm11$ is different from the previous average $\cdots086\pm 22$ by about five error bars. A "five sigma" effect is a big deal in physics, because it is overwhelmingly more likely to be a real physical difference (or a real mistake, ahem) than to be a random statistical fluctuation. This kind of result suggests very strongly that there is physics we misunderstand in the chain of analysis. This is where discoveries come from.
This level of detail becomes important as you try to decide whether the explanations for other puzzles in physics are mundane or exciting. The abstract of the technical paper refers to two puzzles which are impacted by this change: the possibility that a new interaction has been observed in beryllium decays, and the tension between predictions and measurements of the muon’s magnetic moment, which is sensitive to hypothetical new interactions in a sneakier way.
The fine structure constant $\alpha$ pops up all over the place in physics, and its presence in equations that physicists have to solve makes it important to measure its true value, so more accurate predictions can be obtained from those equations.
More than that, in the case of $\alpha$ it is actually the ratio of other fundamental constants of nature whose values are very accurately known. If the calculated value of alpha differed from its measured value, it would hint at something missing from our understanding of the underlying physics- some subtle effect that no one had ever seen before, and which now must be explained. This is the stuff of which Nobel prizes are made, which makes adding significant digits to a measurement like that worthwhile.
I am adding a few thoughts after two very good answers are already there, so I won't repeat material already covered, except to note the element of misreporting in the news item; see answer by rob.
What I want to do is say a little about what it is like to do this kind of physics. I have never myself reported a new high-precision measurement of a fundamental constant, but I have worked with people who have. The classic example of this type of work is the measurement by Lamb and Retherford in 1947 of a tiny effect in hydrogen that is now known as the Lamb shift. It was important because although it only contributes minutely to the energy, it shows that there is an effect in hydrogen that is not predicted by a Dirac equation model of hydrogen; this both stimulated the development of quantum electrodynamics and gave to the theoreticians a definite number to aim at in checking their work.
Ever since then experimental physicists of a certain type have been painstakingly seeking higher and higher precision as a way to probe for new physical effects in the fundamental particle interactions. It is a sort of quiet companion to the particle collision experiments. The latter are a lot more expensive and yield a lot more data, but just occasionally the quiet sort of experiment, working at low energy and ultra-high precision, lends a little nudge. It is very exciting when this happens.
It is not yet clear whether this has happened with this new measurement. The 5 sigma difference means either someone made a mistake, or there is new physics. The mistake could be, for example, that the uncertainty in the old value was underestimated by a factor 2 or 3, and the values just happened to disagree by a few sigma. It is the sort of tantalizing situation that in cosmology would be called a "tension" between differing measurements of ostensibly the same quantity.
(Comment was deleted, so I'm turning it into an answer.)
This 2018 blog post at Résonaances explains some of the physical implications. In brief: the precision with which $\alpha$ is known is a limiting factor in the precision of standard model calculations of the magnetic moments of the electron and muon. The level of precision that both experiment and theory have achieved in determining $g_e$ is such that non-standard-model processes in higher order Feynman diagrams could affect the agreement between experiment and theory.
