AMPK stands for AMP-Activated Protein Kinase. It is a cellular intermediate which is activated by high AMP levels (For those of you new to enzymology, enzymes are typically named exactly for their function; its a great way to use big words that are self explanatory).
AMPK is what is known as a 'Metabolic Lever' (my own term, although credit goes to some others for influence). By focusing on any metabolic lever you can cause a cascade of different reactions that all centralize around a certain goal.
For example, mTOR is an 'anabolic' and 'muscle building' lever when activated and it appears to be a 'longevity' lever when suppressed. Insulin is the lever of glucose metabolism, and noradrenaline seems to be a lever for fat loss.
Levers are typically central to metabolism, and manipulating them holds great promise over the body.
AMPK can be seen as a 'catabolic' lever. It induces beta-oxidation (fat burning), glucose oxidation (carb usage), and enzymes/transporters in both pathways that aid in those two goals (increases enzyme levels that decay glycogen and enzymes that get fatty acids ready for beta-oxidation; primarily hormone-sensitive lipase. it also increases the uptake of both carbs and fats into cells from the blood).
By these mechanisms, AMPK activators can serve as anti-obesity agents.
That being said, in the nature of being a 'catabolic' lever it can also suppress protein synthesis. It does not decay skeletal muscle per se, but it can reduce the efficacy of building it. Mice without AMPK actually build muscle faster and bigger until they get the 'beetus. Downregulating AMPK actually increases the rates of muscle synthesis as well.
So it is powerful; it holds great therapeutic potential, but it isn't perfect. Nothing in the body is.
The exact mechanisms of muscle protein synthesis inhibition are not exactly known, but once a compound is discovered that inhibits AMPK's inhibition of MPS you can be sure it will be the 'next big thing' to be thrown into a supplement stack for 'lean gains'.
Good sources to read regarding AMPK are: