A drink hole is the opening on casino floors that the drink servers come out of and is usually a kitchen area. It is wise to gravitate towards said areas to take advantage of the free drinks.
All you have to do is sit at a slot machine near the drink hole, and look like your gambling. Soon enough, a drink girl will emerge from said drink hole to quench your thirst.
Nick: "All this winning on the slot machines is making me thirsty"
Doug: "In that case, Lets go hit up a drink hole"