You can trim kitty claws, you're just not supposed to trim them down too far. Inside the base of the claw to about halfway along it there is a fleshy colored bit called the quick (the quick can also be a darker color sometimes). The quick is full of blood vessels, so you do not want to accidentally nick it. If you do, you'll hurt kitty. Generally, when trimming kitty claws, you only want to cut some of the hooked part. Stay away from the wider part of the claw. Also, if you go to trim your cat's claws, they will get annoyed. It's best to have some treats nearby and maybe give them 1 or 2 after every couple claws.
My cat had a problem where his claws were a bit too durable and they wouldn't just naturally break off with use. This would lead to the claws growing too long and curling back into his paws if left unchecked. I had to trim his claws every few weeks tot keep that from happening. Following the above rules, I regularly trimmed my kitty's claws for all 17 years of his life without hurting him a single time.