I think the websites you mention talk about two different things: the now extinct meal voucher program, and the Disney Dining Experience that is currently on special.
The 'new' program simply gives you 'meal credits' on your room key - Table Service, Counter Service, and Snack. There are lists floating around with the most recent Table Service/Counter Service locations, and there's new hoopla over snacks being a lot more fun to redeem.
To redeem a counter service credit, you go to one of the listed counter service locations, order an entree, a dessert, and a non-alcoholic beverage (or a combo meal) and give them your room card. They deduct one counter service credit from your pool. There are few things that you can't get, and they're all discussed here, so browse a bit.
To redeem a table service credit, MAKE RESERVATIONS. You go to a restaurant and order an appetizer, entree, dessert, and non-alcoholic beverage (or a buffet) and give them your room card. They deduct one table service credit from your pool, and tip the waitstaff 18% of your comped meal total. There are also exceptions here: if an appetizer or dessert says it's "for two," you normally have to be using two table service credits, and you only get that one dessert/appetizer instead of the normal two. The other exceptions are discussed in detail as well. You also have the option of going to a particularly nice (or fun) restaurant (Signature Dining Experiences) and ordering the same combination but trading in two table service credits instead of one.
The bottom line is, that you should only have to pay extra for *very*'extra' things - you don't pay for the $11.99 pasta dish, and you don't pay for the $34.99 steak, shrimp, and ribs, but you'll pay for your alcohol. If you find a menu that shows an upgrade, do a quick search here: these people will know if they charge extra for it =)