How do slats actually work?
A slat is a retractable version of a device known as “leading edge slot”. It is basically a slot just behind the leading edge. 

When an airplane is flying slow, it will probably either be somewhat nose-up to maintain level flight, or be descending, or somewhere in between. In either case, the air strikes the wing at a higher angle than when the plane is facing forward and flying straight at high speed. The angle with which the air hits the wing is called the angle of attack.

On the surface of the wing, there is a layer of slow-moving air called the boundary layer, caused by friction between the wing and air. This boundary layer makes the wing appear thicker than it is, particularly towards the back, since the air piles up toward the back. The faster airflow prefers to flow around this boundary layer, instead of through it/around the foil, meaning the fast-moving air is turned less, and so produces less suction at the top of the foil. 

At the trailing edge, this boundary layer terminates in turbulence, and creates more suction at the back. Obviously, due to the curvature of the airfoil, the backward side of the top of the wing is more angled towards the back and less upwards, so having suction in that region creates more drag and less lift than having suction further forward.

In effect, due to the boundary layer, the airflow around the wing “sees” the top rear portion of the wing bulge out more than it actually is. The resulting “shape” of the wing is flatter at the top, creating less lift, and has a blunter, steeper trailing edge, creating more drag.

At the higher angle of attack mentioned above, the boundary layer over the top of the wing is even thicker, since, as you can guess, the airflow around the wing would be slowed more on the more difficult journey. Eventually, at very back, the direction of airflow in the boundary layer may reverse, (this is what is called flow separation) creating even more turbulence and increasing suction in that region even more, making something that is bad worse.   

With increasing angle of attack, the boundary layer continues to thicken, and the zone of separated flow expands forward, eventually there will come a time when the wing “bulges” out so much it no longer produces an increase in lift, but produces a lot of drag. This is a stall.

The slot behind the extended slat lets in more fast moving air, speeding up and thus thinning the boundary layer.

So, what is the difference between flaps and slats? 


Flaps deflect air already flowing past the wing downward. Slats enable fresh air to flow over the wing. (NOTE: there are flaps with slots in them. These slots work the same way as slats.)
Flaps can work at zero angle of attack. Slats can't.


Also, just a clarification for the OP about the “droop”. Leading edge droop is exactly what it sounds like. It gives the air a smoother journey over the wing at high angle of attack, increasing lift. But at low angles, it produces negative lift.....