You simply purchase a refillable mug in your resort's food court. You'll be able to find them easily. There are usually a couple of displays with them around the checkouts.
Once you purchase the mug, you are allowed to refill them as many times as you want throughout your stay. There are some limitations on what you can get as a refill. If I am remembering correctly, you might not can get juices and milk. I think everything else is acceptable. Someone at your resort can give you a rundown on what you can get.
We always drive and are able to bring along extra water, milk and colas so we never buy the mugs. We also have a refrigerator in our room (stay at the CBR) to keep our fresh fruit, left over desserts, etc.
If you aren't planning on hanging around the resort much, you might not want to buy a mug. It just depends on how many times you plan on going to the food court. I believe the mugs are now approximately $12.00 so you'd just have to do the math on whether or not it would be cost efficient for you buys.