There are different models with different characteristics/features. There is also the Revlock brand of auto-clutches.
Rekluse probably has a better explanation of how it works on their website, but here is the gist of how my bike works with my model (Z-Start Pro):
At idle or any RPM below the engagement RPM (which you can set) the clutch is disengaged even though the bike is in gear and you do not pull in the lever (the lever can be retained, or you can do away with it altogether - you can still use the lever, but on my model the pull is *very* light).
When you increase the RPM the clutch starts to engage. Where it engages and how hard or soft is up to how you setup the clutch (I think the Revlock Dyna-Ring lets you adjust this on the fly). Once the clutch is fully engaged it stays engaged until the RPM falls below the engagement RPM. If you let off the throttle the clutch stays engaged until the RPM falls below the engagement RPM - so you still have engine compression braking on deceleration or down a hill. A lot of people think you don't, but they obviously have never ridden a bike with a Rekluse because in that respect it is just like a regular clutch.
Here is where the auto-clutches shine: if you are climbing a hill in too high a gear and you lug the engine down, the clutch will slip enough to keep the engine running. Rather than you having to moderate the clutch to keep the engine running, you moderate only the throttle and possibly the gear shift. If you come to a stop, the clutch will slip enough to keep the engine running. No more stalling. No more fanning the clutch or otherwise moderating it and the throttle at the same time. No more having the bike jump all over the place because you let out the clutch too fast or too much. You just concentrate on the bike and the throttle and let the clutch do its work.
Downsides:
No bump starting (Dyna-Ring supports bump starting).
When you let completely off the throttle at a stop, the clutch disengages, which means it can roll backwards or forwards (especially if the engine is off).
The clutch can drag (depending on how you have it setup) when in gear at idle or the engine off, making it a little harder to roll the bike backwards - such as when turning around in a tight spot.
You have to remember - in gear + engine running + throttle = forward momentum. When trying to get over something from a stop this can mean the bike moves forward when you aren't paying attention - so pay attention. But it is still easier than using the clutch lever.
Some people don't like working without a clutch lever, or the much lighter feel. Most don't miss it at all and replace the clutch lever with a LHRB - I did. Only on the street do I miss the lever - where I shift a lot more and want a smoother shift.