All credits for your party/room are stored on all room keys/keys to the kingdom cards. The same card which holds your park tickets, and which opens the door to your resort room.
To use credits, you only need to present the cashier/waiter one of the cards, and all applicable credits will be applied to your bill. In most cases, this will zero out your tab and you will be on your way.
At TS restaurants, you will be asked to sign a slip which resembles a credit card charge slip to acknowledge the use of your credits. Anything extra, like adult beverages or appetizers or such, will be due and can be paid with cash, credit card, room charge, etc. You can handle the gratuity in similar fashion, either paying it in cash or charging it to a credit card or your room charge account.
At CS restaurants, the cashier will scan one of your cards and hand you a receipt which shows the credits used and the credits remaining. If you have extra items, or items not covered by the credits, then the cashier will give you that total and you pay it using cash or credit or room charge.
Snack redemption works just like CS redemption, with the cashier presenting you a receipt showing credits used and credits remaining for your party's cumulative account.
Any member of your party/room can present their card to cover everyone's meals. It only requires one card to take care of the tab. Many people who have room charging privileges on the adults' cards will present a child's card (without room charging privileges) when using dining credits to avoid having the meal be mistakenly charged to the room charge account instead of the dining credits. We've never had problems with credits or charging errors, but some folks have and it sounds like a good careful way to avoid it if you are concerned.