Coffee grease is actually a little bit like the foam of beer, right, the grease is actually a foam, it's a bunch of very small, dense bubbles.  The bubble is mostly carbon dioxide.  The source of this carbon dioxide is roasted coffee beans, which continue to release carbon dioxide, more quickly when ground to increase the surface area of the grains. As the carbon dioxide is released, the beans become less fresh.  

The deeper the beans are roasted, the more carbon dioxide is released, so some lightly roasted beans may produce coffee with little or no oil, but that doesn't mean the coffee is bad!  Of course, if the beans are roasted too deep, and the beans are almost charred, there won't be much carbon dioxide, and there won't be any nice fat.  

The same goes for coffee extraction, where hot water passes through the ground coffee and exerts pressure, dissolving a lot of carbon dioxide.  When the extraction is complete, the coffee is exposed to normal temperature and pressure, and the carbon dioxide that was dissolved in the water starts to release out in large quantities, forming bubbles.  It's not all fat that surrounds these bubbles. In fact, there's very little fat in them. It's mostly polysaccharides, protein, and some very fine coffee grounds.  They form these little bubbles that are very dense, and that's what you call grease.