Once an application has been installed into the state channel, the multisignature wallet has transferred control over the installed amount from the free balance to the application's resolve function, a mapping from application state to funds distribution. For example, in the case of Tic-Tac-Toe, a possible payout function is: if X wins, Alice gets 2 ETH, else if O wins Bob gets 2 ETH, else send 1 ETH to Alice and Bob.
As the underlying state of the application changes, the result of the payout function changes. It is the job of the Update Protocol to mutate this state, independently of the rest of the counterfactual structure.
Using our Tic-Tac-Toe example, if Alice decides to place an X on the board, Alice would run the Update Protocol, transitioning our state to what is represented by the figure above. Notice how both the board changes and the local nonce for the app is bumped from 0 to 1. To play out the game, we can continuously run the update protocol, making one move at a time.