The bladed scarf says that you can cause damage to an opponent with it if your opponent succeeds in a grapple check. However... if the check is successful, you're grappled, and grappled characters can't take an action that requires two hands to perform. The bladed scarf is a two-handed weapon.
If I want to enter a grapple, and I'm wielding a bladed scarf, what happens?
If you're wielding the bladed scarf and use the grapple action, make a grapple check (at -4 for not having two hands free). If successful, you are grappling the target, just as if you had tried to grapple holding some other object. You cannot use the scarf to deal damage when successfully maintaining the grapple, though; you'll have to settle for unarmed strike, natural attack, or armor spike damage instead.
When your opponent makes a successful grapple check against you, however, he takes damage. This presumably includes grapple checks to escape or reverse the grapple.
Also scarf dancers can use them one-handed
In such a case, the grappler could use the scarf to deal damage when maintaining the grapple just like any other one-handed weapon.