Short answer:
- The rotor is a symmetrical airfoil with the chord at right angles to the radius.
- An incoming airstream at a range of angles of attack will "attach" to the airfoil at the rounded leading edge and leave at the sharp trailing edge.
- The air leaves straight back along the chord. If it also came in parallel to the chord you just get drag. But for other angles of attack you've also bent its path, changing its momentum and applying a reaction thrust to the blade.
- The change in momentum is transferred to the blade, appearing as "lift" on the curving front portion. For a significant range of angles of attack the lift force exceeds the drag force and you get net mechanical power.
- Split this lift into radial and circumferential components: The radial component stresses the blade against its structure and the circumferential component pulls it forward.
- The range of angles of attack where the air attaches is small and the range where you get net gain is still smaller. So when the rotor is stopped there are only a narrow range of positions where it will find thrust to attempt to start it. Thus Darrieus rotors are usually not self-starting.
- But the airfoil "flies in the APPARENT wind", the vector sum of the real wind with the wind from its own motion. Once it's spun up - to a tip speed significantly above the wind speed - it "sees" the wind as coming from a direction that oscillates around the direction of its chord. The air stays attached and the rotor gets net power from the wind through most of its rotation, far more than it loses when the blades are running nearly straight into or away from the wind and it experiences more drag than forward lift.
The downside to Darrieus designs is that they are somewhat lacy and hard to support against the radial forces, the radial forces cycle in direction with every turn (fatiguing the material), and the rotation is very fast so the centripital forces are also very strong. This tends to lead to mills that come apart violently in a storm after a few years, vibrate a lot, and otherwise are hard to make reliable.
Upside is a Darrieus (like a HAWT) can get pretty close to the Betz limit. But a Sandia Savonius gets to about 2/3 of Betz, has a tip speed somewhat less than the wind rather than maybe 6 or 7 times as fast, and is MUCH easier to make big and strong so scaling it up by 50% to match a Savonius might be easier. Thus that, or structurally stronger hybrid designs such as the Lenz, are more popular with some of the WAWT-fans on this board.