Normally, they mount the wheel over a hub on the balancer and then screw down this cone to center the rim and tighten it down. That's what doesn't work on these rims. The lug centric adapter fits on their balancer. It has lugs on it to fit the rim. The rim/tire then mount to that adapter and then it is tightened down. The adapter is to properly center these rims on the balancer. The adapter usually has multiple holes to fit different lug patterns for different rims.
I've had three different places use the lug centric adapter. And I've found a couple that had no clue what I was talking about.
If that doesn't answer your question, please click the envelope icon above this post and email me. I'll see that more quickly than a post here.
-- on edit -- the following site is for Toyota vehicles but near the bottom it talks about lug centric balancing and hub centric balancing.
http://www.leadfreewheels.org/101.shtml
-- on 2nd edit -- this is pasted from another site and might explain better than I did.
The basic idea is to mount the wheel on the balancer the same way it is mounted on the vehicle. A pilot hole centric wheel (one where the center hole positions the wheel on the hub and prevents it from wobbling sideways when the lug nuts are removed) can be mounted on a balancer with a cone from the backside. But a lug centric wheel (one that does have some sideways movement when the lug nuts are removed) requires a different balancer mounting procedure. A lug centric wheel should be mounted with a cone from the backside and an adapter flange plate against the front side. The fingers on the flange plate must be properly positioned so they line up with the lug holes in the wheel. This is necessary to center the wheel on the WHEEL balancer shaft. If this is not done, the results will not be 100 percent accurate or repeatable.