After an address is accepted with the Local Delivery module, the message is queued to the Module queue. Each Module process takes messages from the queue, and opens the addressed Account.
If the Account has some Automated Rules specified, these Rules are applied: for each rule its conditions are checked, and if they are met, the specified Rule actions are performed. As a result of those actions, the message can be copied to some Mailbox, a copy of the message can be redirected to some other addresses, an automatic reply can be generated, etc.
You can use a more detailed Log Level for the Local Delivery module to see which Rules are applied to messages, why some conditions are not met, and what actions have been taken when all Rule conditions have been met.