In The Three-Body Problem, Liu Cixin describes the orbit of Trisolaris (Proxima b?) as chaotic. It's a major aspect of the plot. And this article references Barnes et al. (2016) for the hypothesis that "a passage of Proxima close to α Cen may have destabilized the original orbit(s) of Proxima’s planet(s)". That's pretty cool!
Summary: Proxima Centauri really does orbit Centauri A & B (which orbit each other as a binary system). Before now it was suspected that Proxima Centauri orbited the A/B pair, but the data wasn't completely convincing (it could have just been swinging by, since it's "orbit" takes 1/2 million years).
There are some details in the article. Primarily insights about composition of all three stars stars the stability of formation of planets.
I'm not an astronomer. But i'd encourage you to think about where Jupiter is right now. Where will it be in an hour? Where will it be in 10,000 years? It's not particularly important to know where Jupiter is. It doesn't really affect tides, or seasons. But fine discrepancies in Jupiter's orbit can only be explained with relativity. Careful measurement leads to great forecasts, new theories, new science.
Now we know where Proxima Centauri will be in 550,000 years, and where it was 550,000 years ago. Right where it is now. Because it's actually in orbit. It's not a stray merrily zipping by, like some asteroid flying past earth. It's locked into orbit with the other two. It's like our moon.
It's probably not a big deal, just another note on some map. But people gather up tens of thousands of those notes, look for little discrepancies, and find new science. Which i think is pretty cool.
Great way to put it, especially the last paragraph. They say that the most exciting phrase in science is "hey, that's funny...". Sometimes you have to gather a lot of data before that can happen.
jfoutz has a pretty good answer, but as a former astronomer who studied triple systems I thought I would expand a little bit. The main reason this is important is that it is data in favor of the hypothesis that Proxima Centauri is gravitationally bound to the rest of the Alpha Centauri system. If the Alpha Centauri system is triple system rather than a binary with another stars that just happens to be passing by, this could change our understanding of how this system formed.
As it turns out, the dynamics of triple systems can in some cases be extremely interesting due to a phenomenon called Kozai-Lidov oscillations [1]. The idea here is that if the orbital plane of a tertiary companion is at high inclination relative to the inner binary, it can drive the orbit of the inner binary to high eccentricities over a long period of time.
In this case, it is known that Proxima Centauri hosts a planet in its habitable zone. But the orbital dynamics of this planet may have been affected by the Alpha Centauri A/B binary. If the planet formed further away from Proxima, it may have migrated due to Kozai-Lidov oscillations. Furthermore, if Proxima is bound, its dynamical influence on the Alpha Cen A/B system may have prevented the formation of circumbinary planets, or disrupted the orbits of these planets. Indeed, one speculative idea is that Proxima's planet may have formed around Alpha Cen A/B, and then been kicked out and captured by Proxima Cen.
