Thank god Ham radio is still in existence... RF engineering is, essentially, black magic and Hams are the temple acolytes.
As a side note: take a look at how much garbage got posted before people with knowledge stepped in to correct it. It's a small point on something like this, but I think it's generally true that ignorance spreads faster than knowledge. Some of it is random sampling-- there are a small number of experts so it will take time for one to find the article and care enough to post-- and I think some of it is that it takes time to compose a knowledgable response but two keystrokes to type "BS".
The product is quite possibly complete bunk, but that doesn't make the statements of why it's bunk true.
To mostly reinforce what others have said:
You can affect received signal strength without physically touching the existing antenna through
parasitic elements and reflectors. Radio waves are not like electrical current-- they radiate across open space and don't need a closed circuit. Metal and non-metal objects can affect the field through conductivity or their dielectric effects.
There are many dimensions of antenna performance including efficiency, directivity and tuning. It's unlikely that this is affecting the real part of efficiency, though it may be affecting the tuning. I'd hope Apple tuned their antenna well, but we've seen before that different people hold their phones differently and how you hold it can change the tuning.
Most likely, this is trying to do exactly what Arn's post shows: change the pattern of the antenna to add strength in specific directions. While some point out that this will help only when the increased gain is aimed at the tower, the flip side is that it won't hurt if the lost gain is in the direction of your head. The tradeoff will basically be that you have more power directed towards the tower or maybe something that will reflect it towards the tower, and less power making your brain warm.
And the argument that, if it worked, Apple would have done it is not entirely solid. While Apple puts a lot of effort into antenna engineering, they're constrained by cost and industrial design. If these elements require the stand off spacing provided by the case, Apple may have been unwilling to add the size or manufacturing cost. It may also be found that Apple tests the phone in one user-held configuration and these guys test it in another.
So I don't have a whole lot of faith that a few parasitic elements added to a case will change the world, but that doesn't mean that such things are physically impossible.