I have yet to visit the French Laundry, but I have used that ticketing system twice now.
Basically, you're putting down money in advance for your meal. It's really just a guarantee for the restaurant that people show up when they make a reservation. Think of it like attending the theater or a sporting event. You pay for a ticket, and your seat is ready when you arrive.
As for what each restaurant offers, it varies from place to place. Some offer a prix fixe menu. When you're done with the meal, you get up and go. No one even brings a check to the table. The meal and gratuity are inclusive of the price.
Other places apply the ticket cost as a credit toward your meal. In those situations it's more like a deposit, and there's more flexibility as far as your meal goes.
I wasn't really sure about this idea at first, but I honestly kind of like it. It makes things easy. And it's better than arriving at the restaurant and finding you can't get in because they've overbooked it.