"I'm all thumbs." = I am very clumsy with my hands and can't seem to do anything complicated with them. I seem to have grown a set of thumbs - means the same thing.

I don't watch the show, so I couldn't tell you for sure what this means in context. But in English, when someone says they are "all thumbs" they mean to say that they are clumsy. So this person was saying that they felt very clumsy with their hands, maybe like the feeling when your hand feels funny because you laid on it in your sleep. Hope this helps!