The demon slayer skirt is a piece of armour specially tuned for fighting demons. It is dropped by Kal'gerion demons and a demon boss as part of a Demon Flash Mob, which will show up at a number of places every full hour. There will be a message in your chatbox describing the location of the Demon Flash Mob when you are level 120 combat or higher. This message can be toggled off at the Doomsayer in Lumbridge.
Wearing the demon slayer skirt while killing demons on a slayer assignment will give a 1% increase to Slayerexperience gained during the task. Equipping a full set of demon slayer armour will give an 8% total increase to Slayer experience gained while on the task. Wearing the skirt also grants a 2% experience boost in the chosen combat style, as well as 4% damage boost against demons.