Autophagy is a repurposing of the same cell machinery that is used to engulf and digest large pieces of external material. Instead of extending a membrane outward from the cell surface to engulf an external particle, it forms a membrane inside the cell and uses it to engulf some part of the cell's contents. Either way, the result is a membrane-bound bag of stuff, which is then merged with the cell's lysosome, which is full of acid and digestive enzymes.
Autophagy is a fairly recently-discovered mechanism (compared to, say, apoptosis), but it already seems to be a universally-used pathway, for everything from killing invading bacteria, to clearing plaques and aggregates of misfolded proteins (such as those that cause neurodegenerative diseases), to self-cannibalizing unnecessary cell components for energy and building blocks during starvation conditions.
There's some evidence that autophagy is protective against a wide range of neurodegenerative diseases, so a good bit of research is focused on how to stimulate autophagy in neurons in a safe and controlled fashion (since obviously too much autophagy will kill the cell outright).
I'm not super familiar with the current literature on autophagy. I'm a bioinformatician, and I collaborated on an autophagy-related project a few years ago, so I had to read up on it then. I only really recall the overarching themes at this point. But I do remember that since autophagy is used for so many disparate purposes, and because it could easily kill the cell if it got out of control, the regulatory machinery controlling it is almost certainly quite complex.
However, that paper says nothing about neurons, yet it seems to the basis of most of the "Could excercise stimulate neuroprotective autophagy?" speculation that I can find via Google. So I don't know that there's any published link between excercise and neuronal autophagy. I suspect that it's just the paper above showing a link between excercise and muscular autophagy, combined with the untested hypothesis of "Maybe excercise stimulates autophagy in other tisues too."
By the way, note that while the mentioned study was published in January 2012, it was submitted in September of 2010, which means all the research described in the paper happened before then. I wouldn't be surprised if the current research by that same lab was focusing on establishing a link to autophagy in neurons.
