In theory, the cat emery board is a clever idea. Cats like to scratch things, and typically their owners try to redirect that instinct towards their toys and away from the furniture. The sharper a cat's claws are, the more damage they can do with their scratching (both to things and people). So this company decided, why not let the cats scratch on an emery cat board? They can trim their own nails while enjoying themselves.
A nice idea -- but does it work?
If your cat is willing to use the cat emery board, there's no reason it shouldn't work as described. The board's makers have put a lot of effort into making this an appealing cat toy as well as a shortcut to trimming cat claws.
The board has a wonderful ergonomic shape, and I don't know anyone whose cat hasn't enjoyed ergonomically shaped scratching pads (which is to say, pads that conform to the cat's natural shape instead of lying flat). It also contains a number of toys and bits of fluff that seem to appeal to many cats, and if none of that works, the makers suggest using catnip to draw the cat in.
If you have a cat, you probably already know this: different cats like different things. Some cats might love that squeaky mouse while others try to bury it in the litter box. Unfortunately, the same holds true for something like the emery cat board: some cats will like the toys it comes with, some won't.
Also, some cats don't seem to like the texture. They prefer cardboard (or, as you know, carpet). The cat emery board might be a nice alternative way to trim cat claws, but only if they actually use it.
One final concern: the emery cat board is an actual emery board, which means that if your cat enjoys it enough to really go at it the way it's supposed to, it could possibly cause some damage to the cat. I don't think this is a likely possibility, but it's worth being aware of and checking with your vet.