You need at least four times the influence in the area compared to them. If you have at least the first level of intelligence on them, you can highlight their planets and see exactly what the influence ratio is (or highlight one of your own nearby planets and make sure it's under 0.25). If you are over 4.0, they have a chance to flip. If not, a starbase will help boost you to that goal.

Don't expect them to flip right away, though. They could, but they just as easily may not. Once you reach the flipping threshhold, it is a random chance each turn. I've seen a planet sit for an entire game year without flipping.

You need to get the Korx planets above 4. Getting your planets below 0.25 is just a way of estimating how close you are to getting the Korx planets to 4 if you don't yet have the espionage levels to tell you exactly what the Korx planets are at.

Another tip, don't try to split the difference when building an influence base. Build it touching one of the target planets, as the effect of the base drops much faster with distance than it does from planets. Once one flips the other will follow without any additional effort.

I think he means, if you have 2 planets, don't build the influence base in the middle between the two. Instead build the influence base as close as possible to one planet. Once that one flips, the other planet will follow soon.

Thanks for the tip.  Glad I posted the pic

On the higher AI levels I assume this isn't as easy?  I've taken 11 planets so far without firing a shot.  Almost seems like I'm cheating. 

The rules for flipping don't change with difficulty level, at least as far as I've ever noticed. What will change is how the AI reacts to you building influence bases near their planets - relations will rapidly decline and occasionally they will declare war on you over it.