I guess I can try and add a bit more to this since I'm not sure how direct your questions have been adressed yet. As you can tell from the responses, this is a complex concept that many degreed mets can even struggle breaking down. So, your confusion on the topic is certainly understandable. Let me try taking another stab at it.
Consider your looking at say a 500 mb height map of height contours, which recall is kind of like looking at pressure contours, and as is typical for the northern hemisphere assume it is colder to the north - which would mean lower heights to the north. Let's add in a small 'trough', or shortwave/cyclonic vorticity maxima at this height, and assume that there is no such feature near the surface - so only consider what is happening above. Recall that at mid-levels the winds blow roughly parallel to height contours, and faster as the contours get closer together. So in our example we would start with a uniform westerly wind, except where our trough is, as there the speed of the wind is greater where the height contours are closer together - so this is a little jet streak there. Now, height contours at mid-levels are a reflection of how warm or cold the air below it is, so for our example, lower heights further north implies colder mean air temperature further north. So, for our trough to be where it is, the air just north of the 'dipped' contour must be cold relative to points east and west. Visually, our example looks like the image below:
The green blob is our wind maxima - with cyclonic vertical vorticity on the north side of this wind maxima (purple blob). Now, if we wanted to move our shortwave/vorticity maxima to some new location, such as the dashed height contour, AND WE DON"T WANT TO SIMPLY BLOW IT THERE, how could we replicate that same pattern at the new location? Well, if we warmed up the air below the current trough location, the heights would rise and we could 'fill in' the current trough location, and if we cooled down the new trough location enough we could lower heights there to match the new pattern. In the atmosphere, if we don't advect warm/cool air horizontally to cool/warm the air at a given point, we can lift (for dry air we call it adabatic cooling) or sink (subsidence warming) air to get the job done.
In this explaination, vorticity didn't appear to do any of the work - because vorticity is just a kinematic wind property, a tool to try and understand the atmosphere, and not always the easiest way to think about things for all. That said, if we used the above example - we can say our cyclonic vorticity is a maxima at the center of the magenta blob - and since it is embedded in westerly flow, picking a point on the eastern edge of the maxima is seeing increasing cyclonic vorticity from advection - so rising motion. A point in the back side is seeing decreasing cyclonic vertical vorticity, so sinking motion.
As for the other aspect of your question - why anti-cyclonic vorticity doesn't lead to rising motion, well it depends on the scale of the motion. For a circulation that is small, say the size of a thunderstorm (anti-cyclonic supercell perhaps), then cyclonic and anti-cyclonic vorticity play by more equal rules. However, as you get to larger scales in time and space, Coriolis force starts becoming a major player in wind behavior, and the relationships between the centrifugal, Coriolis and pressure gradient forces that control the wind are quite different for cyclonic and anti-cyclonic circulations that essentially acts as a limiter to how 'tight' an anti-cyclonic circulation can get, whereas there is no such limiter on cyclonic circulations. Again, another tough concept to tackle in simple forms.
Don't know if I helped or made it more complex - but good luck with trying to figure things out.
