You do not need to bring your fragments with you to offer them. They can be in your bank and when you offer, it will check your inventory and bank for fragments and award experience accordingly removing any fragments from both locations. Gathering a chronicle fragment gives Hunter experience equal to 10 times your Hunter level. The experience is affected by experience boosts such as the enhanced yaktwee stick and bonus experience (however, the Divination experience gained by turning in fragments is not affected).