The idea is that when a game finishes, it then enters an 'After-Game Chat' phase. This phase lasts for 7 days, after which the game just reverts to a normal finished game status. While in the After-Game Chat phase, anyone who played in the game will still have access to the Shoutbox, and can post there as desired. Players may want to congratulate the winner, discuss parts of the game or whatever they wish. Once the After-Game Chat phase finishes, the Shoutbox will become read-only in the normal way for a Finished game. Note that access to the Shoutbox and the identity of the players in the game will all be available regardless of the variant played - even if the game was a Gunboat or anonymous game. Also, the map will be the normal finished map so in a FoW game it will be the unfogged map.
As far as the user interface is concerned, a game in After-Game Chat will go on the Finished Games list just as other finished games do. However, it will have a status of 'After-Game Chat' and the button that normally says 'View game History' will actually say 'Ignore Chat Phase'. The screenshot below shows these differences.
Notice that the After-Game Chat is in RED. If After-Game Chat is in Red as it is here, that means there is an unread message in the Shoutbox from one of the other players. Once you go in to the game and look at the new Shoutbox messages, it will go back to the normal colour until another message is posted. This is to help you see someone else has commented in the chat. The colour always starts off as Red when you finish your game just to let you know the after-chat is now open.
The 'Ignore Chat Phase' button is for you to signal that you no longer want to be part of the After-Game Chat. Click this, and the display will go back to that of a normal finished game. From now on when you look at your Finished Games list, this game will now look just like any other finished game.
I hope I explained that properly

As usual, I will wait a bit before I put the changes up to give people time to get used to the idea.