Cost relates to the total cost of Servants combined with their currently equipped Craft Essences that can be in a single Party at once. Cost increases as your Master Level increases, with a cap of 109 total cost at Master Level 120.
The more stars a card has, the higher its cost will be. Exceptions to this rule include Mashu Kyrielite and limited availability Craft Essences like The Beginning of a Journey and Happy x3 Order. Mashu has a cost of 0 and the limited Craft Essences have a cost of 5 or less, unlike other 4★ Craft Essences.
Support Servants and their equipped Craft Essences have no cost for use in your quest party.