Logic resources are the "guts" of an AGI game. They are scripts that contain all the commands necessary to control the view, picture and sound resources and to receive and act on input from the player, thus determining everything that happens in the game. Each room generally has its own logic which determines what happens in that room, and the game also has a few other logics to look after things like initialization and error handling.
Games are programmed in a special language specific to AGI and compiled into logic resources. The logic programming language is similar to standard programming languages like C, but a lot less powerful as it is designed for specific purpose. For more information on logic programming, click here.
You can edit logics with the Logic editor.