Resource extraction from asteroids is something that has been studied for decades. There is a great article and info-graphic at: Space.com that shows different ideas, from both NASA and Planetary Resources (a company currently planning to mine a near-Earth asteroid). I very much encourage you to read it!
One of the easiest and most useful things to mine would be water, which can be "mined" from an asteroid that has chunks of ice on it. Someday, the water will be valuable for its own sake (for future inhabitants of space stations or lunar colonies, whatever, to drink!) but in the nearer term, the ice would be melted, then split into hydrogen and oxygen (using electrolysis). Hydrogen and oxygen are used to fuel cryogenic rocket engines, and having a bunch of rocket fuel already out in Earth orbit somewhere would be a huge boon for sending large probes to the outer planets, fueling up manned missions to Mars, etc.
More advanced mining ideas would be mining metals for use in automated construction/fabrication in space; I'm sure mining in microgravity is going to have its share of technical challenges, but your question about anchoring a probe or machine to the surface has a bunch of answers, some of which I can think of: use powered darts that fire from the probe during approach and which embed themselves in the asteroid (much like a piton used in climbing). Alternatively, throw a net around the entire asteroid (assuming a small one), and use the net to keep the probe tight against the asteroid to allow drilling the first holes for anchors. Lots of ways to tackle that problem.